Isotropic Markov semigroups on ultra-metric spaces∗ Alexander Bendikov†
Alexander Grigor’yan‡
Christophe Pittet§
Wolfgang Woess¶ Dedicated to the memories of V.S. Vladimirov (1923–2012) and M.H. Taibleson (1929–2004)
arXiv:1304.6271v2 [math.PR] 12 Jul 2013
Abstract Let (X, d) be a locally compact separable ultra-metric space. Given a reference measure µ on X and a step length distribution σ on [0 , ∞), we construct a symmetric Markov semigroup {P t }t≥0 acting in L2 (X, µ). Let {Xt } be the corresponding Markov process. We obtain upper and lower bounds of its transition density and its Green function, give a transience criterion, estimate its moments and describe the Markov generator L and its spectrum which is pure point. In the particular case when X = Qnp , where Qp is the field of p-adic numbers, our construction recovers the Taibleson Laplacian (spectral multiplier), and we can also apply our theory to the study of the Vladimirov Laplacian which is closely related to the concept of p-adic Quantum Mechanics. Even in this well established setting, several of our results are new. We also elaborate the relation between our processes and Kigami’s jump processes on the boundary of a tree which are induced by a random walk. In conclusion, we provide specific examples concering the interplay between the fractional derivative and random walks.
Contents 1 Introduction
2
2 Heat semigroup and heat kernel
10
3 Spectral distribution function
13
4 Basic estimates of the heat kernel
17
5 Green function and transience
22
6 Moments of the Markov process
26
7 The Markov generator and its spectrum
32
8 The p-adic fractional derivative
42
9 Product spaces and the Vladimirov and Taibleson Laplacians
49
10 Random walks on a tree and jump processes on its boundary
57
11 The duality of random walks on trees and isotropic processes on their boundaries
67
12 Random walk associated with p-adic fractional derivative
78
∗
Version of July 01, 2013. Mathematics Subject Classification: 05C05, 47S10, 60J25, 81Q10 Supported by the Polish Government Scientific Research Fund, Grant 2012/05/B/ST 1/00613 ‡ Supported by SFB 701 of German Research Council § Supported by the CNRS ¶ Supported by Austrian Science Fund projects FWF W1230-N13 and FWF P24028-N18 †
1
1
Introduction
In the past three decades there has been an increasing interest to various constructions of Markov chains on ultra-metric spaces, such as Cantor set or the field of p-adic numbers. We introduce and study a class of symmetric Markov semigroups and their generators on ultra-metric spaces. Our construction is very transparent and has a geometric appeal. It leads to a number of new results as well as to a better understanding of previously known results. Let (X, d) be a metric space. The metric d is called an ultra-metric if it satisfies the ultrametric inequality d(x, y) ≤ max{d(x, z), d(z, y)}, (1.1) that is obviously stronger than the usual triangle inequality. In this case (X, d) is called an ultra-metric space. The ultra-metric property (1.1) implies that the balls in an ultra-metric space (X, d) look very differently from Euclidean balls. Any two closed ultra-metric balls of the same radius are either disjoint or identical. The minimal radius of a ball coincides with its diameter. It follows that the collection of all balls of the same radius r forms a partition of X. This can be used to characterize the distance d (x, y) between points x, y ∈ X as follows: d(x, y) equals to the minimal value of r such that both x and y belong to the same ball of radius r. This property in turn implies that to define an ultra-metric d on X one has to prescribe to each ball B the value d(B) of its diameter; see §10. One of the best known examples of an ultra-metric space is the field Qp of p-adic numbers endowed with the p-adic norm kxkp and the p-adic ultra-metric d(x, y) = kx − ykp . Moreover, for any integer n ≥ 1, the p-adic n-space Qnp = Qp × ... × Qp is also an ultra-metric space with the ultra-metric dn (x, y) defined as dn (x, y) = max{d(x1 , y1 ), ..., d(xn , yn )}. All ultra-metric spaces considered in this paper are assumed to satisfy two natural conditions: (i) (X, d) is complete and separable. (ii) Every closed ball Br (x) = {y ∈ X : d(x, y) ≤ r} is compact. If in addition (X, d) satisfies the third condition: (iii) The group of isometries of (X, d) acts transitively on X, then (X, d) is in fact a locally compact Abelian group, that can be identified with any Abelian subgroup of its group of isometries that acts transitively on (X, d). For example, this is the case for Qnp . Observe that this identification is not unique (!). With an eye on typical applications, in the course of our study we pay specific attention to the following two basic cases: Case 1.1 (X, d) is discrete and infinite. Case 1.2 (X, d) is perfect, that is, it contains no isolated point.
2
Various constructions of Markov processes on non-compact perfect ultra-metric spaces (X, d) satisfying the hypotheses (i) – (iii) have been developed by Evans [22], Haran [27], [28], Ismagilov [31], Kochubei [36], [37], Albeverio and Karwowski [2], [1], Albeverio and ´ -Talamanca [39], [40], Rodrigues-Vega and Zuniga-Galindo Zhao [3], Del Muto and Figa [63], [47]. They fixed any identification of (X, d) with a locally compact Abelian group (in fact, Qp ) and studied X-valued infinitely divisible random variables and processes by using tools of Fourier analysis; for general references, see Hewitt and Ross [29],Taibleson [52] and Kochubei [37]. Indeed, Taibleson’s spectral multipliers on Qnp are early forerunners of the Laplacians that we are considering here. Condition (iii) has later been relaxed but still all constructions relied crucially on an identification of (X, d) with some metric space of sequences. See also Pearson and Bellissard [42] and Kigami [34], [35], where (X, d) is the Cantor set, resp. the Cantor set minus one point. In [34], [35], a main focus is on the interplay between random walks on trees and jump process on their boundaries. In this context, we also mention Aldous and Evans [4] and Chen, Fukushima and Ying [18]. We shall come back to Kigami’s work in the last three sections of this paper. An entirely different approach was developed by Vladimirov, Volovich and Zelenov [54], [56]. They were concerned with p-adic analysis (Bruhat distributions, Fourier transform etc.) related to the concept of p-adic Quantum Mechanics, and introduced a class of pseudo-differential operators on Qp and partially on Qnp . In particular, they considered the p-adic Laplacian defined odinger equation. Among other results, they on Q3p and studied the corresponding p-adic Schr¨ explicitly computed (as series expansions) certain heat kernels as well as the Green function of the p-adic Laplacian. In connection with the theory of pseudo-differential operators on general totally disconnected groups we mention here the pioneering work of Saloff-Coste [48]. Ultra-metric spaces (X, d) that satisfy conditions (i)–(ii) and which are discrete were treated by Bendikov, Grigor’yan and Pittet [7], the direct forerunner of the present work. Among the examples of such spaces we mention the class of locally finite groups: a countable group G is locally finite if any of its finite subsets generates a finite subgroup. Every locally finite group G is the union of an increasing sequence of finite subgroups {Gn }. An ultra-metric d in G can be defined as follows: d(x, y) is the minimal value of n such that x and y belong to a common coset of Gn . Let us list a few well known examples of locally finite groups. • Z(p) ⊕ Z(p) ⊕ · · · = lim Z(p) ⊕ · · · ⊕ Z(p) – the infinite sum of the cyclic groups Z(p); −→ | {z } n times
• S∞ := lim Sn – the infinite symmetric group; −→
• SL∞ (p) := lim SLn (p) - the infinite special linear group over the finite field Z(p) . −→
Here are further interesting examples of ultra-metric spaces. • Boundaries, resp. punctured boundaries of locally finite trees - these are the generic examples, since every locally compact ultra-metric space can be realized in this way. (Compare with §10 below.) • Profinite groups, that is, groups which are inverse limits of finite groups. They are compact and totally disconnected; see Ribes and Zalesskii [46]. If such a group G has a firstcountable topology then that topology is generated by an ultra-metric: first-countability 3
amounts to the existence of a decreasing sequence of open normal subgroups Nk with trivial intersection. If x, y ∈ G are distinct then there is a minimal k such that xNk 6= yNk , and d(x, y) = 2−k defines a suitable ultra-metric.According to the Levy-Malˇcev theorem any Abelian profinite group A appears as the intersection of the center Z(G) and the commutant [G, G] of some compact group G, and the mapping G → A is onto; see Hofmann and Morris [30]. • For any positive integer N , the collection of all isomorphism classes of finite or infinite connected, rooted graphs with vertex degrees bounded by N is well-known to be a compact ultra-metric space. An ultra-metric is given as follows: the distance between two nonisomorphic graphs (resp. representatives) is 1/r, where r is the largest integer such that the balls of radius r around the root (with respect to the discrete graph metric) are isomorphic. • A closed subspace of the last one is given by the collection of all finite or infinite rooted trees with vertex degrees bounded by N . We briefly return to our discussion of locally finite groups. Since they are not finitely generated, the basic notions of geometric group theory such as the word metric, volume growth, isoperimetric inequalities, etc. (cf. e.g. Gromov [26]), do not have their counter-part in this setting. Hence one cannot apply directly the well-developed methods to study random walks on finitely generated groups provided by Varopoulos, Saloff-Coste and Coulhon [53], Coulhon, Grigor’yan and Pittet [19], Woess [58], Pittet and Saloff-Coste [43], [44], [45], Saloff-Coste [49], and others. Additional arguments are needed. The notion of an ultrametric can be used instead of the word metric in this setting. See [7] and as well as Bendikov, Bobikau and Pittet [6], [5]. Selecting a set of generators for each subgroup Gn , one defines thereby a random walk on Gn , that is, a Markov kernel on Gn . Taking a convex combination of the Markov kernels across all Gn , one obtains a Markov kernel on G that determines a random walk on G = ¨ s [20], Kesten and lim Gn . Such random walks have been studied by Darling and Erdo −→
Spitzer [33], Flatto and Pitt [24], Fereig and Molchanov [23], Kasymdzhanova [32], Cartwright [16], Lawler [38], Brofferio and Woess [14], see also Bendikov and SaloffCoste [9]. In particular, [38] has a remarkable general criterion of recurrence of such random walks. The main novelty here is that many of those results are subsumed by our approach via ultra-metrics: we develop tools to analyse a class of very natural and simply defined Markov processes on ultra-metric spaces (discrete and non-discrete as well) without assuming any group structure, but only under the hypotheses (i) and (ii). We do not need to use any identification of the ultra-metric space (X, d) with a set of sequences. In particular, the nature of our arguments becomes completely geometric-analytic and allows us to bring into consideration an arbitrary Radon measure µ on X (instead of the Haar measure in the case of groups), that will be used as a speed measure for a Markov process. This approach has certain historical reminiscences: the theory of elliptic operators in Rn can be developed first in the case of constant coefficients by using the Fourier transform, whereas the case of variable coefficients requires more advanced tools. The measure µ in our case (see below) will play the role of a variable coefficient. The aforementioned construction of random walks on locally finite groups can be easily generalized to an arbitrary ultra-metric measure space (X, d, µ) : instead of a sequence of subgroups
4
{Gn } one uses the (countable !) family of balls {Br (x) : r > 0, x ∈ X}, and in each ball one has to choose a Markov kernel. The latter will be chosen as the µ-uniform distribution in each ball. Here is our construction. In addition to the measure µ on X, we fix a probability measure σ on the half-line [0 , ∞) with σ({0}) = 0 and define the following Markov operators, where r ≥ 0 and f ∈ L∞ (X, µ). Z 1 f dµ, (1.2) Qr f (x) = µ Br (x) Br (x) Z ∞ Qr f (x) dσ(r). (1.3) P f (x) = 0
It can be proved that Qr is an orthoprojector in L2 (X, µ), which implies that P is a bounded self-adjoint operator in this space. Moreover, P is non-negative definite, which allows us to define the powers P t for all t ≥ 0. Then P t t≥0 is a symmetric strongly continuous Markov semigroup. In fact, P t for t > 0 admits the following representation: Z ∞ t Qr f (x) dσ t (r) , (1.4) P f (x) = 0
t where σ t is the probability measure on [0, ∞) defined by σ t [0 , r) = σ[0 , r) , and P 0 = id . The semigroup P t determines a continuous time Markov process {Xt }t≥0 . At the same time, one can consider a discrete time Markov chain {Xn }, n = 0, 1, ... with the transition operator P . It has the following transition rule: firstly, a radius r is chosen at random according to the probability σ, then Xn+1 is µ-uniformly distributed in Br (Xn ). Since the n-step transition operator of this Markov chain is P n , we see that the discrete time Markov chain coincides with the restriction of the continuous time Markov process {Xt } to integer values of t. This allows us to study both processes simultaneously, although we will consider only the continuous time process {Xt }t≥0 . Definition 1.3 We refer to the Markov semigroup P t t≥0 as defined by (1.2)-(1.4) as an isotropic semigroup, and to the process {Xt }t≥0 – as an isotropic jump process.
Let us emphasize that the definition of P t (and {Xt }) depends on the following three data: the ultra-metric d, the reference measure µ, and the distance distribution σ, so that we can refer to P t and {Xt } as (d, µ, σ)-semigroup resp. process. There is some degree of freedom, though. For example, one can modify simultaneously d and σ in a certain way, while keeping µ, so that P t does not change.
Definition 1.4 The standard process associated with µ and d is the one where σ = σ ∗ is the “inverse exponential distribution” whose distribution function is σ ∗ ([0 , r)) = e−1/r , r > 0 .
(1.5)
The intrinsic ultra-metric associated with {P t } on X is given by 1 1 . = log d∗ (x, y) σ 0 , d(x, y) 5
(1.6)
Observe that the ultra-metric d∗ induces the same collection of balls as d, and our process becomes the standard (d∗ , µ)-process, as one easily verifies by the corresponding substitution in the integral appearing in (1.3), resp. (1.4). An isotropic random walk on an ultra-metric measure space has a unique feature: its Markov operator P admits a spectral resolution in L2 (X, µ) where the spectral projectors are again Markov operators. This property brings up a new insight, new technical possibilities, and a new type of results, that have no analogue in other commonly used settings. For example, in our setting, the composition of any increasing function with the generator of the Markov semigroup is again a Markov generator. This property is in striking contrast with the classical result of Bochner about general Markov generators, where one needs to compose with a Bernstein function. In the context of random walks on groups, compare with Bendikov and SaloffCoste [11] as well as [10]. In the present study the following issues will be specifically dealt with. (A) Explicit construction of the heat kernel (=transition probabilities) and the spectral distribution function (in §2 and §3). (B) Estimates of the heat kernel and the Green function (in §4 and §5). (C) In particular, criteria of recurrence and transience, including an analogue of Lawler’s criterion without assuming any group structure (in §5). (D) Estimates of escape rates (in §6). (E) A precise description of the infinitesimal generator (= −L , where L is the “Laplacian”) and its spectral properties (in §7). (F) In particular, subordination techniques (also in §7): as it should be clear from the above remark, this will look entirely different from the classical setting. Besides several examples that accompany the elaboration of our theory, we clarify with particular emphasis how that theory is related with two seemingly different analytic, resp. stochastic topics: (G) The p-adic Laplace operator of Vladimirov (as well as related operators) on Qnp is itself not the generator of an isotropic jump process, but it can be treated as the sum of such generators on each of the n coordinates. On the other hand, Taibleson’s Laplacians (spectral multipliers) on Qnp do correspond to isotropic processes. (See §8, §9 and §12.) (H) The processes on the boundary of a tree that arise via “harmonic transform” from random walks on a tree, and that were recently examined by Kigami, embed naturally in our setting. (See §10, §11 and §12.) Let us return to a slightly more detailed outline of the analytic body of this work. One of our aims is to obtain estimates of the transition density p (t, x, y) of the process {Xt } (that is, the integral kernel of P t ) via the probability measure σ and the intrinsic volume function V (x, r) = µ Br∗ (x) , (1.7) 6
where Br∗ (x) is the closed r-ball in the intrinsic metric d∗ . As indicated above, our approach is based upon the observation that the building blocks of the operator P , namely, the averaging operators Qr of (1.2), are orthogonal projectors in L2 (X, µ). This very specific property is a consequence of the ultra-metric inequality (1.1). The range of Qr coincides with the linear subspace Vr of L2 (X, µ) of all functions that are constant on each ball of radius r. Moreover, the spaces Vr are decreasing in r, so that the family {Qr } is (up to reparametrization) a spectral resolution of the identity. Hence, the ultra-metric property leads to the spectral resolution (Eτ ) of P as a self-adjoint operator in L2 , where the spectral projectors are also Markov operators. This enables us to engage at an early stage the methods of spectral theory and functional calculus. Many of our results are stated in terms of the spectral distribution function N defined as N (x, τ ) = 1/V (x, 1/τ ). In the particular case of a discrete space X, we have N (x, τ ) = (Eτ 1{x} , 1{x} ). If the group of isometries of (X, d) acts transitively on X, then N (x, τ ) does not depend on x and coincides with the von Neumann trace of the orthoprojector Eτ . In general, N (x, τ ) is a non-decreasing staircase function on [0, ∞) whose behavior as τ → 0 is intimately related to the behavior of the heat kernel p (t, x, y) as t → ∞ . Indeed, we have the following explicit identities p(t, x, y) = t
Z
1/d∗ (x,y)
N (x, τ ) exp(−τ t) dτ ,
(1.8)
0
p(t, x, x) =
Z
∞
exp(−τ t) dN (x, τ )
(1.9)
0
(Lebesgue-Stieltjes integral with respect to τ 7→ N (x, τ )). Then we obtain our heat kernel estimates using the properties of the Laplace transform. For example, we prove that 1 t N x, p (t, x, y) ≃ t + d∗ (x, y) t + d∗ (x, y) if and only if the function τ 7→ N (x, τ ) satisfies the doubling property (for the latter, see Definition 4.2 below). The relation ≃ means, that the quotient of the left and right hand sides is bounded above and below by positive constants. In particular, if N (x, τ ) ≃ τ α then t
p(t, x, y) ≃
t2 + d∗ (x, y)2
1+α , 2
that is, p (t, x, y) behaves like the Cauchy distribution in “α-dimensional” space. This example is closely related to studying p-adic fractional derivatives of order α. The Green function estimates, as well as conditions for transience of the Markov process {Xt }, are based on the fact that the Green function (more generally, the λ-Green function) is a Stieltjes-type transform of the spectral distribution function N . Using general properties of the Stieltjes transform, we obtain various results. For instance, the process is transient if and only if Z dτ N (x, τ ) 2 < ∞. τ 0 In particular, when N (x, τ ) ≃ τ α as τ → 0, transience is equivalent to α > 1.
7
The identity (1.8) allows us to estimate the moments Mγ (x, t) = Ex (d∗ (x, Xt )γ ) of the process {Xt }. Indeed, we have the following explicit identity Z ∞ t Rγ x, e−τ dτ , Mγ (x, t) = τ 0 where
1 Rγ (x, τ ) = V (x, τ )
Z
sγ dV (x, s) (0,τ ]
(Lebesgue-Stieltjes integral with respect to the intrinsic volume measure s 7→ V (x, s)). In particular, we obtain the following general results: (1) Assume that (X, d) is non-compact, has no isolated points, and that 0 < γ < 1. Then Mγ (x, t) ≤
tγ . 1−γ
Thus in general the moment of order γ is finite for all 0 < γ < 1. If V (x, r) satisfies the reverse doubling condition – see Definition 4.9 below – then there is a matching lower bound. In this case the γ-moment is finite if and only if 0 < γ < 1. (2) Assume that (X, d) is discrete and infinite, and that 0 < γ < 1. Then Mγ (x, t) ≤
c min{t, tγ } 1−γ
for some c > 0. If V (x, r) satisfies the reverse doubling condition then there is a matching lower bound. In this case the γ-moment is finite if and only if 0 < γ < 1. (3) Assume that (X, d) is compact and perfect. then C t, if γ > 1, 1 γ = 1, Mγ (x, t) ≤ C t (log t + 1), if γ Ct , if 0 < γ < 1,
for some C > 0 and all 0 < t ≤ 1. If V (x, r) satisfies the reverse doubling condition then there is again a matching lower bound. If −L is the infinitesimal generator of the (d, µ, σ)-process then in our terminology, L is the Laplacian. We mention the following property: for any continuous, strictly increasing function ϕ : [0 , ∞) → [0 , ∞) such that ϕ(0) = 0, the operator φ (L) is also a Laplacian of an analogously constructed Markov semigroup. In particular, Lα is a Laplacian for any α > 0. Recall for comparison that, for a general symmetric Markov generator −L, the operator −Lα generates a Markov semigroup only for 0 < α ≤ 1. Three further remarkable properties of L are: • L extends to Lp (X, µ), and its spectrum in that space is the same for all p ∈ [1 , +∞). • Assume that (X, d) is non-compact. Let M ⊆ [0 , ∞) be a closed set that accumulates at 0 and such that M is unbounded if X contains at least one non-isolated point. Then there exist a proper ultra-metric d′ on X which generates the same topology and an isotropic jump process on (X, d′ ) whose associated Laplacian L has M as its spectrum.
8
• L has the strong Liouville property, that is, any non-negative L-harmonic function must be constant. We are able to apply our results to locally finite groups, but with arbitrary reference measure µ instead of the Haar (=counting) measure. Some of the aforementioned questions are particularly sensitive to the choice of the speed measure, for example, the heat kernel and Green function estimates. The spectrum of the Laplacian and escape rate bounds do not depend on the speed measure µ. These quantities depend strongly on the choice of the ultra-metric d which generates the topology of X, whereas the eigenfunctions depend on d and µ. Let us now return to outlines of the two major applications of our theory, as indicated above in items (G) and (H). First of all, we show that the p-adic Laplace operator on Qnp introduced by Vladimirov and his collaborators can be studied within our setting, although that Laplacian itself is not the generator of an isotropic jump process. As an important step towards this goal, we first relate our approach to the theory of p-adic fractional derivatives, as introduced by Vladimirov and collaborators [56]. This implies simple direct proofs of many results of [56], without using Fourier Analysis and the theory of Bruhat distributions. On the other hand, on Qnp we also have the Taibleson Laplacian, which is the n-dimensional analogue of the p-adic fractional derivative: we show that this one is the generator of an isotropic jump process, which allows to obtain very detailed analytic results. The other main application has a slightly different flavour. We first outline a structuretheoretical feature, namely, the known fact that every locally compact ultra-metric space arises as the boundary of a locally finite tree. In particular, if the space is compact, then we can describe it as the boundary of a rooted tree. Discrete time Random walks of nearest neighbour type on the vertex set of a tree are very well understood. See Chapter 9 of the book by Woess [60]. It is then natural to ask if there is a connection between such random walks on trees and isotropic jump processes of the form (1.4) on the boundary of a tree. Indeed, in recent work, Kigami [34] starts with a transient nearest neighbour random walk on a tree and constructs a naturally associated jump process on the boundary of the tree. Using this approach, he undertakes a detailed analysis of the process on the boundary. Restricting attention to the compact case, in the next section we shall answer the obvious question how the approaches of Kigami and of the present paper are related: the relation is basically one-to-one. As in [34], we take the ultra-metric space to be the geometric boundary at infinity of a locally finite rooted tree where each vertex has at least two forward neighbors. “Basically” means that we need to restrict to random walks on trees which are Dirichlet regular, that is, the Dirichlet problem at infinity admits solution, or equivalently, the Green kernel of the random walk vanishes at infinity. (We comment on this condition, which appears to be natural in the present context, at the end.) Given such a nearest neighbour random walk on a tree, its reversibility leads to an interpretation of the tree as an infinite electric network. This comes along with a natural Dirichlet form on the tree, and a natural approach is to use the Dirichlet form on the boundary which reproduces the power (“energy”) of harmonic functions on the tree via their boundary values. This form on the boundary is computed with some effort in [34]; it induces the jump process studied there. Now, that form on the boundary is an integral with respect to the Na¨ım kernel, 9
which goes back to the work of Na¨ım [41] and Doob [21] in the setting of abstract potential theory on spaces which are locally Euclidean. Trees do not have the latter property, but the validity of the resulting formula for the power of harmonic functions is proved for general infinite electric networks (≡ reversible random walks) in a forthcoming paper of Kaimanovich and Georgakopoulos [25]. A direct and rather simple proof for the case of trees is also given in §10. We relate the Dirichlet form on the space of harmonic functions with finite power with the Dirichlet form on the boundary that is computed via the Na¨ım kernel in terms of the boundary values of the involved harmonic functions. This leads to an explanation of the relation between our isotropic jump processes of and the processes of [34] induced on the boundary of a tree by a random walk on that tree: every boundary process induced by a random walk is an isotropic jump process of our setting. Conversely, we show that up to a unique linear time change, every isotropic jump process on the boundary of a tree arises from a uniquely determined random walk as the process of [34]. In addition, we explain how the boundary processes on a tree transforms into an isotropic jump process on the non-compact ultra-metric space given by a punctured boundary of the tree. This should be compared with the very recent work [35]. Finally, in §12 we work out in a specific example how the applications of (G) and (H) are related: we consider the p-adic fractional derivative on the (compact) group of p-adic integers and the corresponding random walk on the associated rooted tree, as well as the random walk corresponding to the fractional derivative on the whole of Qp .
2
Heat semigroup and heat kernel
We always consider a proper ultra-metric space (X, d) with the properties (i), (ii) outlined in the Introduction, and let µ be a Radon measure supported by the whole of X. We assume that • µ({x}) = 0 if x ∈ X is not an isolated point. • µ({x}) > 0 if x ∈ X is an isolated point.
(2.1)
• µ(X) = ∞ if X is non-compact. We can also work without the third assumption, but it holds in most typical cases. We set [ Λ(x) = Λd (x) = {d(x, y) : y ∈ X} and Λ = Λd = Λ(x). x
Each set Λ(x) is countable. There is a strictly increasing sequence of numbers rk = rk (x) > 0 such that • if (X, d) is non-compact and x is not isolated then Λ(x) = {0} ∪ {rk : k ∈ Z} with lim rk = ∞ and lim rk = 0; k→∞
k→−∞
• if (X, d) is non-compact and x is isolated then with lim rk = ∞;
Λ(x) = {0} ∪ {rk : k ∈ Z+ }
• if (X, d) is compact and x is not isolated then with lim rk = 0.
Λ(x) = {0} ∪ {rk : k ∈ Z− }
• if (X, d) is compact and x is isolated then is a finite set, with n = nx .
Λ(x) = {0} ∪ {r1 , . . . , rn }
k→∞
k→−∞
The following notion will be used several times in later sections. 10
(2.2)
Definition 2.1 Given a closed ball B = Br (x) ( X, its predecessor is the ball B ′ = Br′ (x), where r ′ is the minimal element of Λ(x) that satisfies r ′ > r. Regarding the distance distribution σ on the half-line that appears in the definition (1.3) of P , we shall always denote its (left-continuous) distribution function by σ(r) = σ [0 , r) and assume that it has the following non-degeneracy properties. σ(0+) = 0 , and σ(r) is strictly increasing and < 1 on each Λ(x) , x ∈ X.
(2.3)
The assumptions (2.3) imply that σ(r) > 0 for every r > 0 and that σ charges every interval [r , r + ε), where ε > 0 and r ∈ Λ. Recall that σ t denotes the probability distribution with distribution function σ(r)t . Also recall the operator Qr from (1.2), Z 1 f dµ. Qr f (x) = µ Br (x) Br (x) Since
Br (x) = Br (y) for every y ∈ Br (x),
(2.4)
we have that Qr f takes constant values on any ball Br (x) with given radius r > 0. In particular, x 7→ Qr f (x) is a bounded continuous function. We can write Qr f in the form Z Kr (x, y)f (y)dµ(y), Qr f (x) = X
where the integral kernel is given by the equation Kr (x, y) = By (2.4), the kernel is symmetric:
1 1Br (x) (y). µ Br (x)
(2.5)
Kr (x, y) = Kr (y, x). Since Qr f ≥ 0 when f ≥ 0, and Qr 1 = 1, the operator Qr is symmetric and Markovian. It follows that QBr f extends as a bounded self-adjoint operator acting on L2 (X, µ) with norm kQr kL2 →L2 = 1. Moreover, for any r, r ′ > 0, Qr′ Qr = Qr Qr′ = Qmax{r,r′ } .
(2.6)
Thus {Qr }r≥0 is a non-increasing family of ortho-projectors. We have Q0 = id. By monotonicity, the strong limit Q∞ = limr→∞ Qr exists on L2 and is again an ortho-projector. If (X, d) is noncompact, it is easy to see that Q∞ = 0. If (X, d) is compact then Q∞ coincides with the orthoprojector on the one dimensional subspace of L2 consisting of constant functions. Consequently, the family of ortho-projectors {Eτ }+∞ −∞ , defined as QB1/τ if τ ≥ 0 Eτ = 0 if τ < 0 defines a spectral measure dEτ in L2 . What is crucial in our analysis is that the measure dEτ gives the spectral resolution of the Markov semigroup (P t ) defined in the Introduction in (1.4). We now verify its basic properties. 11
Theorem 2.2 (1) The family {P t }t≥0 defined in (1.4) is a symmetric Markov semigroup on L2 (X, µ). (2) Under the assumptions (2.3) on σ, the semigroup is strongly continuous. Proof. For any given s, t > 0 and f ∈ L2 , we have by (2.6) Z ∞ Z ∞ dσ t (r ′ ) Qr Qr′ f (x) = dσ s (r) P s P t f (x) = Z0 ∞ Z0 ∞ s dσ t (r ′ ) Qmax{r,r′ } f (x). dσ (r) = 0
0
Let ξ 1 and ξ 2 be two independent random variables with distributions σ s and σ t , respectively. Then the distribution of the random variable ξ = max{ξ 1 , ξ 2 } is σ t+s . It follows that Z ∞ Qr f (x) dσ t+s (r) = P t+s f (x). P s P t f (x) = E Qmax{ξ1 ,ξ2 } f (x) = 0
This proves the first statement. For the second statement, we already observed that Qr f → f strongly as r → 0. Fix ε > 0 and choose δ 1 > 0 such that ε sup kQr f − f kL2 < . 2 0≤r 0 for all r > 0, we can choose δ 2 > 0 such that ε sup kf kL2 (1 − σ t (δ 2 ) < . 4 00}
=
Z
X
Z
1 µ Br (x)
X
!
1Br (x) f (y) dµ(y)
dσ t (r)
! 1 f (y) dσ t (r) dµ(y) , µ Br (x)
{r≥d(x,y)}
which concludes our computation.
3
Z
Spectral distribution function
We continue to work with the (d, µ, σ)-process and the associated symmetric Markov semigroup {P t }t>0 on our proper ultra-metric space X, always assuming that µ and σ satisfy (2.1) and (2.3). Recall the intrinsic ultra-metric defined in (1.6), that is, d∗ (x, y) = −1/ log σ(r) whenever d(x, y) = r. Since σ(r) < 1 by (2.3), this is well-defined. Clearly, the value sets of the new metric are [ Λ∗ (x) = {d∗ (x, y) : y ∈ X} = {−1/ log σ(r) : r ∈ Λ(x)} , and Λ∗ = Λ∗ (x) . (3.1) x
The following is straightforward. Lemma 3.1 For r ∈ Λ(x), resp. r ∗ ∈ Λ∗ (x), we have Br∗∗ (x) = Br (x) ⇐⇒
1 1 , = log ∗ r σ(r)
where Br∗∗ (x) is the closed d∗ -ball of radius r ∗ centred at x. In particular, the families of closed d-balls and of closed d∗ -balls coincide. Definition 3.2 For any x ∈ X we define the spectral distribution function N : [0 , ∞) → [0 , ∞) as ∗ N (x, τ ) = 1 µ B1/τ (x) . See Figures 1, 2 and 3 for qualitative pictures.
13
N(x,τ )
1
µ(X)
τ
0
Figure 1: The graph of the function τ 7→ N (x, τ ) in the case when (X, d) is compact
1
µ(x)
N(x,τ )
τ
0
Figure 2: The graph of the function τ 7→ N (x, τ ) in the case, when (X, d) is discrete and infinite
N(x,τ )
τ
0
Figure 3: The graph of the function τ 7→ N (x, τ ) in the case when (X, d) is not discrete and non-compact
14
Theorem 3.3 For x, y ∈ X, p(t, x, y) = t
Z
1/d∗ (x,y)
N (x, τ ) exp(−τ t) dτ . 0
In particular, p(t, x, x) =
Z
exp(−τ t) dN (x, τ ) [0,∞)
(Lebesgue integral with respect to the measure whose distribution function is N (x, ·)) Proof. Let Λ(x) = {rk } as in (2.2), where k ranges in Z, Z+ or Z− , or the range of k is finite according to the respective case. Observe that on each interval [rk , rk+1 ), we have Br (x) = Brk (x). Thus, the function x 7→ µ Br (x) takes constant values on that interval. Let y be such that d(x, y) = rn . By Theorem 2.4 we can write Z XZ dσ t (r) dσ t (r) = p(t, x, y) = µ B (x) r [r ,r ) {r≥rn } µ Br (x) k k+1 k≥n X σ(rk+1 )t − σ(rk )t µ B (x) r k k≥n Z σ(rk+1 ) X 1 = tξ t−1 dξ. µ B (x) r σ(r ) k k k≥n =
Now let
τ (r) = log
1 , σ(r)
0 ≤ r ≤ ∞.
By the assumptions (2.3) on σ, τ (rk+1 ) < τ (rk ) and
τ (rk+1 ) ≤ τ (r) ≤ τ (rk ) for rk ≤ r ≤ rk+1 .
∗ ∗ By Lemma 3.1, with the substitution ξ = e−τ , and using that B1/τ (rk ) (x) = B1/τ (x) for every τ with τ (rk+1 ) < τ < τ (rk ), Z σ(rk+1 ) Z τ (rk ) 1 1 t−1 tξ dξ = t exp(−τ t) dτ ∗ µ Brk (x) σ(rk ) µ B1/τ τ (rk+1 ) (rk ) (x) Z τ (rk ) Z τ (rk ) t exp(−τ t) dτ = t N (x, τ ) exp(−τ t) dτ . = ∗ τ (rk+1 ) τ (rk+1 ) µ B1/τ (x)
Summing over all k ≥ n, the first identity of the theorem follows. The second identity follows from the first one by integration by parts. Let us define t∗ = t∗ (x) ≤ ∞ as follows. t∗ = lim sup τ →∞
log N (x, τ ) . τ
(3.2)
It controls the exponential decay of the intrinsic volume function V (x, r) of (1.7), as r → 0. Corollary 3.4 (a) For any x, y and t > 0, 0 < p(t, x, y) < min{p(t, x, x), p(t, y, y)} ≤ ∞ . (b) p(t, x, x) = ∞ for 0 < t < t∗ , and p(t, x, x) < ∞ for t > t∗ . 15
Proposition 3.5 Assume that (X, d) is compact and that t∗ (x) < ∞ for some x ∈ X. Then uniformly in y ∈ X, 1 . lim p(t, x, y) = t→∞ µ(X) Proof. We have τ 0 > 0, where 1/τ 0 is the d∗ -diameter of X. Hence N (x, τ ) = 1/µ(X) for 0 < τ < τ 0 . We write Z ∞ N (x, τ ) exp(−τ t) dτ = p(t, x, x) = t 0 Z ∞ 1 = 1 − exp(−τ 0 t) + t N (x, τ ) exp(−τ t) dτ . µ(X) τ0
Since t∗ (x) < ∞ , we can choose A, a > 0 such that N (x, τ ) ≤ A exp(aτ ) for all τ > τ 0 . Then for all t > 2a we obtain Z ∞ At exp(−τ 0 (t − a)) t N (x, τ ) exp(−τ t) dτ ≤ t−a τ0 τ0 t ≤ 2A exp − . 2 It follows that lim p(t, x, x) =
t→∞
1 . µ(X)
(3.3)
For arbitrary y ∈ X we know that p(t, x, y) ≤ p(t, x, x),
(3.4)
and t t , x, x ≥ p p(t, x, y) ≥ p(t, x, x) − 2 exp − 2d∗ (x, y) 2 τ 0 t p( 2t , x, x) ≥ p(t, x, x) 1 − 2 exp − . 2 p(t, x, x)
(3.5)
Equations (3.3), (3.4) and (3.5) yield the result. Proposition 3.6 Assume that (X, d) is non-compact and that t∗ (x) < ∞ for some x ∈ X. Then p(t, x, y) lim = 1, t→∞ p(t, x, x) locally uniformly in y. Proof. Choose A, a > 0 such that N (x, τ ) ≤ A exp(aτ ) for all τ > 0. For t > 2a and y ∈ X we write Z ∞ N (x, τ ) exp(−τ t) dτ (3.6) p(t, x, x) − p(t, x, y) = t 1/d∗ (x,y) Z ∞ t−a . exp −τ (t − a) dτ ≤ 2A exp − ≤ At d∗ (x, y) 1/d∗ (x,y)
16
On the other hand p(t, x, x) =
Z
exp(−τ t) dN (x, τ ) ≥
(3.7)
[0 , ∞)
Z
≥
exp(−τ t) dN (x, τ ) ≥ exp −
[0 , 1/2d∗ (x,y))
t 2d∗(x,y)
N
1 x, 2d∗ (x, y)
.
Let K be a compact set and D(K) be its d∗ -diameter. The inequalities 3.6 and 3.7 yield the following: for y ∈ K, t−a 2A exp − p(t, x, x) − p(t, x, y) d∗ (x,y) ≤ t 1 p(t, x, x) exp − 2d∗ (x,y) N x, 2d∗ (x,y) t−2a 2A exp − 2D(K) → 0 as t → ∞. ≤ 1 N x, 2D(K) This completes the proof.
4
Basic estimates of the heat kernel
The purpose of this section is to provide estimates of the heat kernel of an isotropic semigroup as was previously defined. Recall that by Theorem 3.3 p(t, x, y) = t
Z
1/d∗ (x,y)
N (x, τ ) exp(−τ t) dτ .
(4.1)
0
Proposition 4.1 For all x, y ∈ X, the following estimates hold. (a) For all 0 < t ≤ d∗ (x, y), t 1 1 t N x, N x, ≤ p(t, x, y) ≤ . 2e d∗ (x, y) 2d∗ (x, y) d∗ (x, y) d∗ (x, y) (b) For all t > d∗ (x, y), 1 N 2e
1 N e
(c) For all t > 0,
x,
1 2t
1 x, t
≤ p(t, x, y).
≤ p(t, x, x).
Proof. For (a), we use monotonicity of τ 7→ N (x, τ ) and write p(t, x, y) = t
Z
1/d∗ (x,y)
0
p(t, x, y) ≥ t
Z
t N N (x, τ ) exp(−τ t) dτ ≤ d∗ (x, y)
1/d∗ (x,y)
1 x, d∗ (x, y)
and
N (x, τ ) exp(−τ t) dτ 1 t 1 t t N x, )≥ N x, exp(− . 2d∗ (x, y) 2d∗ (x, y) d∗ (x, y) 2e d∗ (x, y) 2d∗ (x, y) 1/2d∗ (x,y)
≥
17
For (b), when 1/d∗ (x, y) > 1/t we obtain p(t, x, y) ≥ t
Z
1/t
N (x, τ ) exp(−τ t) dτ
0
≥ t
Z
1/t
N (x, τ ) exp(−τ t) dτ ≥
1/2t
1 N 2e
x,
1 2t
.
For (c), using the second formula of Theorem 3.3, we write Z ∞ Z ∞ N (x, τ ) exp(−τ t) dτ N (x, τ ) exp(−τ t) dτ ≥ t p(t, x, x) = t 1/t 0 Z ∞ 1 1 1 exp(−τ )dτ = N x, . ≥ N x, t e t 1 The proof is finished. Definition 4.2 A non-decreasing function Φ : R+ → R+ is said to satisfy the doubling property if there exists a constant D > 0 such that Φ(2s) ≤ D · Φ(s) for all s > 0. It is known (Potter’s theorem) that if Φ is doubling then Φ(s2 ) ≤ D
s2 s1
δ
Φ(s1 )
for all 0 < s1 < s2 , where δ = log2 D.
(4.2)
Proposition 4.3 For any given x ∈ X, the following two properties are equivalent. (1) For some constant c > 0 and all t > 0, 1 1 1 N x, ≤ p(t, x, x) ≤ c N x, . c t t (2) The function τ 7→ N (x, τ ) is doubling. Proof. (2) ⇒ (1): The lower bound is the inequality of Proposition 4.1(c), so what is left is to prove the upper bound. We have Z ∞ N (x, τ ) exp(−τ t) dτ p(t, x, x) = t 0 Z ∞ τ exp(−τ ) dτ . = N x, t 0 By assumption, N is doubling, whence Z
∞ 0
τ N x, exp(−τ ) dτ t
Z ∞ N x, τt 1 exp(−τ ) dτ = N x, t N x, 1t 0 Z ∞ 1 ≤ D N x, max{1, τ δ } exp(−τ ) dτ t 0 1 . ≤ D ′ N x, t
18
The constants D, D ′ , δ > 0 come from Potter’s bound (4.2). (1) ⇒ (2): Using the inequality of statement (1), we obtain Z ∞ 1 N (x, τ ) exp(−τ t) dτ c N x, ≥ p(t, x, x) ≥ t t 2/t 2 . ≥ e−2 N x, t Setting τ = 1/t, we come to the desired conclusion. Theorem 4.4 The following inequalities hold for all x, y ∈ X and t > 0. ! 1 t 1 , p(t, x, y) ≥ N x, 2e t + d∗ (x, y) 2 t + d∗ (x, y) t + d∗ (x, y) t p , x, x . p(t, x, y) ≤ 2e t + d∗ (x, y) 2
Proof. When 0 < t ≤ d∗ (x, y) we use Proposition 4.1(a): t 1 1 N x, p(t, x, y) ≥ 2e d∗ (x, y) 2d∗ (x, y) ≥
t 1 1 N x, 2e t + d∗ (x, y) 2 t + d∗ (x, y)
When t > d∗ (x, y) we use Proposition 4.1(b): 1 1 N x, p(t, x, y) ≥ 2e 2t ≥
1 t 1 N x, 2e t + d∗ (x, y) 2 t + d∗ (x, y)
!
.
!
.
Similarly for the upper bound on p(t, x, y) : depending on whether 0 < t ≤ d∗ (x, y), or t > d∗ (x, y), we will have t + d∗ (x, y) t t p , x, x ≥ p d∗ (x, y), x, x t + d∗ (x, y) 2 2d∗ (x, y) 1 t 1 1 ≥ N x, p(t, x, y), ≥ 2e d∗ (x, y) d∗ (x, y) 2e or
t + d∗ (x, y) 1 t p , x, x ≥ p(t, x, y), t + d∗ (x, y) 2 2e
respectively, as desired. Corollary 4.5 Suppose that τ 7→ N (x, τ ) is doubling. Then ct 1 1 Ct N x, N x, ≤ p(t, x, y) ≤ t + d∗ (x, y) t + d∗ (x, y) t + d∗ (x, y) t + d∗ (x, y) for all t > 0, y ∈ X and some C, c > 0. 19
The following three cases are of particular interest. Example 4.6 Assume that (X, d) is non-compact and has no isolated points, e.g. X = Qp . Assume that for a given x ∈ X, and some α, β > 0, α τ , 0 < τ ≤ 1, N (x, τ ) ≃ τ β, τ > 1. Then for all t > 0 and all y ∈ X, ( p(t, x, y) ≃
−1−β , 0 < t + d∗ (x, y) ≤ 1, t t + d∗ (x, y) −1−α , t + d∗ (x, y) > 1. t t + d∗ (x, y)
Example 4.7 Assume that (X, d) is discrete, e.g. X = x ∈ X, N (x, τ ) ≃ τ γ
L∞
k=1 Z(nk ).
Assume that for a given
for some γ > 0 and all 0 < τ ≤ 1. Then for all t > 0 and all y ∈ X such that t + d∗ ≥ 1, p(t, x, y) ≃
t
1+γ . t + d∗ (x, y)
Example 4.8 Assume that (X, d) is compact and has no isolated points, e.g. X = Zp , the set of p-adic integers. Assume that for a given x ∈ X, N (x, τ ) ≃ τ δ , for some δ > 0 and all τ > 1. Then for all t > 0 and all y ∈ X such that 0 < t + d∗ (x, y) ≤ 1, p(t, x, y) ≃
t
1+δ . t + d∗ (x, y)
Thus in all three examples p(t, x, y) has a shape similar to that of the Cauchy density in Euclidean space. The parameters α, β, γ and δ play the role of a local dimension (resp. dimension at infinity) of the space X. Definition 4.9 A non-decreasing function Ψ : R+ → R+ is said to satisfy the reverse doubling property, if Ψ(r) ≥ (1 + η)Ψ(δr), for all r > 0 and some δ, η ∈ (0 , 1). The function is said to satisfy the reverse doubling condition at 0, resp. at ∞, if there is R > 0 such that the above inequality holds for all r with 0 < r < R, resp. all r > R. Note that Ψ satisfies the reverse doubling property if and only if its generalized inverse Ψ−1 is doubling, where Ψ−1 (s) = sup{r : Ψ(r) ≤ s}. We have the following sufficient criterion for N (x, ·) to have the doubling property. Proposition 4.10 The function τ 7→ N (x, τ ) is doubling if the following two conditions hold: (1) The function Ψ(r) = −1/ log σ(r) satisfies the reverse doubling property. 20
(2) The volume function r 7→ V (x, r) = µ Br (x) satisfies the doubling property.
Proof. It follows from the Definition 3.2 of the spectral distribution function that τ 7→ N (x, τ ) is doubling if and only if the function s 7→ µ(Bs∗ (x)) is doubling. By definition, resp. Lemma 3.1, Bs∗ (x) = Br (x), , r = Ψ−1 (s) ,
∗ and B2s (x) = Br′ (x) , r ′−1 (2s).
Condition (2) implies that for some constants A, a > 0, ∗ (x) = µ Br′ (x) ≤ A (r ′ /r)a µ Br (x) = A (r ′ /r)a µ Bs∗ (x) . µ B2s
To estimate the ratio (r ′ /r)a we use condition (1):
1 1 1 1 1 1 1 . log = log ≤ log ≤ ··· ≤ log ′ ′ k 2 σ(r) σ(r ) 1+η σ(δr ) (1 + η) σ(δ k r ′ ) We choose k > 1 such that (1 + η)−k < 1/2 . For such k we obtain log
1 1 < log . σ(r) σ(δ k r ′ )
It follows that r ≥ δ k r ′ , whence ∗ µ (B2s (x) ≤ A δ −ak µ Bs∗ (x) .
The proof is finished.
Example 4.11 Let (X, d) be the field of p-adic numbers Qp equipped with its canonical ultrametric d(x, y) = kx − ykp . Let µ be the additive Haar measure on Qp . Let Zp ⊂ Qp be the group of p-adic integers. For x ∈ Qp , k ∈ Z and pk ≤ r < pk+1 , we have Br (x) = x + p−k Zp In particular, for all r > 0,
and µ Br (x) = pk .
r < µ Br (x) ≤ r. p
(4.3)
Let us choose the distance distribution with distribution function σ(r) = exp −(p/r)α , where 1 = (r/p)α . By Proposition 4.10, the function τ → N (x, τ ) is α > 0. Then Ψ(r) = 1/ log σ(r) doubling. Direct computations show that kx − ykp α 1 d∗ (x, y) = and r α < µ Br∗ (x) ≤ p r 1/α , p whence
1 1/α τ ≤ N (x, τ ) < τ 1/α . p
By Corollary 4.5, there exist constants c1 , c2 > 0 such that for all t > 0 and all x, y ∈ Qp , (t1/α
c2 t c1 t ≤ p(t, x, y) ≤ 1/α . 1+α + kx − ykp ) (t + kx − ykp )1+α
21
β On the other hand, choosing σ(r) = exp − log(1 + 1r ) , where β > 0, we obtain 1 exp(τ 1/β ) − 1 ≤ N (x, τ ) ≤ p exp(τ 1/β ) − 1 . p
Thus, the spectral distribution function τ → N (x, τ ) is not doubling on the whole range of τ ∈ (0, ∞), and Corollary 4.5 does not apply. Note that N (x, τ ) ≃ τ 1/β at 0, whence we can apply Theorem 4.4 and the standard Laplace transform argument to show that p(t, x, y) ≃
t (t1/β + kx − ykp )1+β
when t + kx − ykp ≥ 1,
compare with Proposition 3.6. With other choices of the distance distribution σ, one can consider many other interesting examples.
5
Green function and transience
The resolvent {Rλ }λ>0 associated with our (or any) semigroup {P t }t>0 is defined on the set of bounded Borel measurable functions f as Z ∞ e−λt P t f dt. Rλ f = 0
Since the semigroup admits an integral kernel p(t, x, y) with respect to the reference measure µ, the resolvent admits an integral kernel Gλ (x, y) as well. We call it the λ-Green function. It can be represented in the form Z ∞ Gλ (x, y) = e−λt p(t, x, y) dt . 0
Theorem 5.1 Let {P t } be defined by (1.4). Then its λ-Green function is given by Gλ (x, y) =
Z
1/d∗ (x,y)
0
N (x, τ ) dτ . (τ + λ)2
The following properties hold: (a) The function d∗∗ : X × X → R+ defined as ( 1/Gλ (x, y) if x 6= y d∗∗ (x, y) = 0 if x = y is an ultra-metric on X generating its topology. (b) For any given λ > 0, Gλ (x, y) ∼ λ
Z
−2
1/d∗ (x,y)
N (x, τ ) dτ
as d∗ (x, y) → ∞ .
0
(c) Assume that Gλ (x, x) = ∞. Then Gλ (x, y) ∼
Z
1
1/d∗ (x,y)
N (x, τ ) dτ τ2 22
as d∗ (x, y) → 0 .
(5.1)
Proof. Using the definitions of Rλ and Gλ (x, y), we obtain Z Z ∞Z Gλ (x, y)f (y) dµ(y), e−λt p(t, x, y)f (y) dµ(y) dt = Rλ f (x) = 0
X
X
so that Gλ (x, y) is indeed the integral kernel of the operator Rλ . Integrating the identity (4.1) with respect to t, we obtain (5.1). Let d∗ (x, z) = a and d∗ (y, z) = b, assume that a ≤ b. Then by the ultra-metric inequality, d∗ (x, y) ≤ b. Using the symmetry of the λ-Green function, we write Z 1/a Z 1/b N (z, τ ) dτ N (z, τ ) dτ Gλ (x, z) = ≥ = Gλ (y, z) and 2 (τ + λ) (τ + λ)2 0 0 Z 1/d∗ (x,y) Z 1/b N (y, τ ) dτ N (y, τ ) dτ Gλ (x, y) = ≥ = Gλ (y, z). 2 (τ + λ) (τ + λ)2 0 0 It follows that Gλ (x, y) ≥ Gλ (y, z) = min{Gλ (x, z), Gλ (z, y)}, whence the function d∗∗ (x, y) = 1/Gλ (x, y) satisfies the ultra-metric inequality. For any fixed x ∈ X, the new distance d∗∗ (x, y) is a strictly increasing function of d∗ (x, y). Therefore it generates the same topology as d∗ . This proves (a). Let λ > 0 be fixed and d∗ (x, y) → ∞. Using (5.1), we get Z 1/d∗(x,y) 1 N (x, τ ) dτ Gλ (x, y) ≥ 2 0 1/d∗ (x, y) + λ Z 1/d∗ (x,y) 1 Gλ (x, y) ≤ N (x, τ ) dτ . λ2 0
and
These two inequalities imply the asymptotic behavior proposed in (b). The proof of (c) is analogous. Corollary 5.2 Assume that the function τ 7→ N (x, τ ) is doubling. Then we have the following. (1) If Gλ (x, x) = ∞ then
1 Gλ (x, y) ≃ d∗ (x, y) N x, d∗ (x, y)
as d∗ (x, y) → 0.
(2) If X is non-compact then 1 1 N x, Gλ (x, y) ≃ d∗ (x, y) d∗ (x, y)
as d∗ (x, y) → ∞ .
The potential operator R associated with the semigroup {P t }t>0 is defined on the set of non-negative Borel measurable functions f as Rf (x) = lim Rλ f (x). λ→0
Definition 5.3 (1) The process {Xt }t>0 and the semigroup {P t }t>0 are called transient if the potential Rf is a bounded function whenever f is bounded and has compact support. (2) The Green function G(x, y) is defined as
G(x, y) = lim Gλ (x, y). λ→0
23
Using (5.1), we obtain G(x, y) =
Z
1/d∗ (x,y) 0
N (x, τ ) dτ ≤ ∞. τ2
(5.2)
Obviously, transience requires non-compactness of X, and when X is compact, G(x, y) = ∞ for all x, y ∈ X (compare with Figure 1). Theorem 5.4 Suppose that (X, d) is non-compact. Then the following statements are equivalent. (1) The semigroup {P t }t>0 is transient. (2) G(x, y) < ∞ for some/all distinct x, y ∈ X. (3) For some/all x ∈ X,
Z
1 0
N (x, τ ) dτ < ∞. τ2
Proof. The equivalence of statements (2) and (3) follows from the very definition of the Green function. To prove the equivalence (1) ⇐⇒ (2), set f = 1B , where B = Br∗ (a) with a ∈ X and r ∈ Λ∗ (a), as defined in (3.1). Consider the potential R(x, B) := Rf (x). By the Maximum Principle, sup R(x, B) = sup R(x, B). x∈X
x∈B
Let B ′ = Br′ (a) be the predecessor ball of B in the sense of Definition 2.1. It exists by noncompactness of X. Write Z G(x, y) dµ(y) R(x, B) = =
Z
B
dµ(y) B
Z
1/d∗ (x,y)
0
N (x, τ ) dτ = τ2
Z
(0 , r ′ )
dV (x, s)
Z
1/s
0
N (x, τ ) dτ , τ2
where dV (x, s) refers to Lebesgue-Stieltjes integration with respect to the intrinsic volume function (1.7) at x. Changing the order of integration we obtain ! Z ∞ Z N (x, τ ) R(x, B) = 1(0 , 1/s] (τ ) dV (x, s) dτ τ2 0 (0 , r ′ ) ! Z ∞ Z N (x, τ ) = dV (x, s) dτ . τ2 0 (0 , r ′ )∩(0 , 1/τ ] By Definition 3.2 , N (x, τ )V (x, τ1 ) = 1 for all τ > 0. Hence Z
Z ∞ N (x, τ ) N (x, τ ) 1 R(x, B) = V (x, r) dτ + V (x, ) dτ 2 2 τ τ τ 1/r ′ 0 Z 1/r′ N (x, τ ) = V (x, r) dτ + r ′ . 2 τ 0 1/r ′
24
Observe that r ′ = d∗ (x, z) for any x ∈ B and any z ∈ B ′ \ B. Moreover N (x, τ ) = N (y, τ ) for any x, y ∈ B and τ ≤ 1/r ′ . It follows that for any x ∈ B and any z ∈ B ′ \ B, R(x, B) = r ′ + µ(B)G(a, z). The last identity yields the equivalence of (1) and (2). When X is discrete and satisfies the conditions (i)–(iii) of the Introduction, i.e. the group of isometries acts transitively, then X can be seen as a locally finite group. If µ is the Haar measure, and we consider the discrete time processes, then the transience criterion (3) of the last theorem translates into the the general sufficient transience condition of [38]. Thus, the latter is also the necessary one. Corollary 5.5 Let (X, d) be non-discrete. Assume that the semigroup {P t } is transient and that G(x, x) = ∞ for some (equivalently, all) x. Then (1) For any given λ > 0, as d∗ (x, y) → 0, Gλ (x, y) ∼ G(x, y). (2) Assume in addition that the function τ 7→ N (x, τ ) is doubling. Then, as d∗ (x, y) → 0, 1 G(x, y) ≃ d∗ (x, y) N x, . d∗ (x, y) Consider the transient case. Fix x ∈ X, set d∗ (x, y) = r and ϕ(τ ) = τ N (x, 1/τ ), and write Z ∞ dτ ϕ(τ ) . G(x, y) = τ r A typical result in Karamata theory reads as follows. Z ∞ dτ ϕ(τ ) ≃ ϕ(r) as r → ∞ τ r if and only if the function F satisfies certain Tauberian conditions, see Bingham, Goldie and Teugels [13, Corollary 2.6.4]. In our setting, this means that 1 , as d∗ (x, y) → ∞ G(x, y) ≃ d∗ (x, y) N x, d∗ (x, y) if and only if the function τ 7→ τ N (x, 1/τ ) satisfies the above mentioned Tauberian conditions. A simple sufficient condition for the last property to hold is given in the next theorem. Theorem 5.6 Assume that there exist constants 0 < ε′′ < ε′ < ε < 1 such that ε′′ ≤
N (x, ετ ) ≤ ε′ N (x, τ )
(5.3)
for some x ∈ X and all 0 < τ ≤ 1. Then the semigroup {P t }t>0 is transient and, for all y ∈ X such that d∗ (x, y) ≥ 1, 1 . G(x, y) ≃ d∗ (x, y) N x, d∗ (x, y) 25
Proof. For any 0 < η ≤ 1 we have Z
0
η
∞
X N (x, τ ) dτ = τ2
Z
εk η
k+1 η k=0 ε
∞
X N (x, τ ) dτ = τ2 k=0
Z
η εη
N (x, εk τ ) −k ε dτ . τ2
Using the upper bound in (5.3), we obtain Z
0
η
N (x, τ ) dτ τ2
≤
∞ Z X k=0
=
η
εη
N (x, τ ) ′ k −k 1 1−ε ηN (x, η) (ε ) ε dτ ≤ 2 ′ τ 1 − ε /ε (εη)2
N (x, η) 1 − ε N (x, η) = c2 . ′ ε(ε − ε ) η η
Analogously, using the lower bound in (5.3) we get Z η Z η N (x, τ ) N (x, τ ) 1 − ε N (x, εη) dτ ≥ dτ ≥ 2 2 τ τ ε η 0 εη ′′ ε (1 − ε) N (x, η) N (x, η) ≥ = c1 . ε η η Setting η = 1/d∗ (x, y) and using (5.2), we come to the desired conclusion. Example 5.7 This is a continuation of Example 4.11: X = Qp is the field of p-adic numbers with standard distance d(x, y) = kx − ykp and with modified distance d∗ (x, y) = p−α kx − ykαp , α > 0. The spectral distribution function N (x, τ ) does not depend on x, whence we denote it N (τ ). It satisfies p−1 τ 1/α ≤ N (τ ) ≤ τ 1/α . By Theorem 5.4, the semigroup {P t }t>0 is transient if and only if α < 1. Moreover, for all x, y, G(x, y) ≃ kx − yk−1+α . p Note that in this example (and in many others) the condition (5.3) is in fact equivalent to transience. Indeed, for any fixed 0 < ε < 1 and all τ > 0, ε1/α ≤
N (ετ ) ≤ p ε1/α . N (τ )
When α < 1, we can choose 0 < ε < p−α/(1−α) , ε′ = p ε1/α and ε′′ = ε1/α to show that the condition (5.3) holds. When α ≥ 1, ε ≤ ε1/α ≤
N (ετ ) , N (τ )
whence the condition (5.3) does not hold.
6
Moments of the Markov process
Let {Xt } be the Markov process associated with the semigroup {Pt }t>0 . For any γ > 0, the moment of order γ of the process is defined as Mγ (x, t) = Ex d∗ (x, Xt )γ , 26
where Ex is expectation with respect to the probability measure on the trajectory space of {Xt } that governs the process starting at x. In terms of the transition function p(t, x, y), that moment is given by Z d∗ (x, y)γ p(t, x, y) dµ(y).
Mγ (x, t) =
(6.1)
X
The aim of this section is to estimate Mγ (x, t) as a function of t and γ. For the main result, we shall need two lemmas. Recall the intrinsic volume function (1.7). We also need its average moment function of order γ at x, Z 1 Rγ (x, τ ) = r γ dV (x, r). V (x, τ ) (0,τ ] Lemma 6.1 For all x ∈ X, t > 0 and γ > 0, Z ∞ Z ∞ 1 t −τ t Rγ x, Mγ (x, t) = Rγ x, te dτ = e−τ dτ . τ τ 0 0
Proof. Using the equations (6.1) and (4.1), as well as the Definition 3.2 of the spectral distribution function in terms of the volume function, we obtain Z d∗ (x, y)γ p(t, x, y) dµ(y) Mγ (x, t) = X ! Z Z rγ
=
1/r
t
=
Z
∞ 0
N (x, τ ) e−τ t dτ
dV (x, r)
0
(0 , ∞)
Z
(0 , 1/τ )
! Z ∞ rγ 1 −τ t Rγ x, dV (x, r) t e dτ = t e−τ t dτ . V (x, 1/τ ) τ 0
In the 3rd identity, we have used Fubini’s theorem. The volume function r 7→ V (x, r) non-decreasing. In view of its relation with the spectral distribution function (see Definition 3.2) it is a step function whose shape can be understood from figures 1 – 3. The function varies from 0 to µ(X). In the compact case, V (x, r) = µ(X) ∗ (x), the largest value in Λ (x); compare with (2.2) and (3.1). When x is ∗ = rmax for all r ≥ rmax ∗ ∗ ∗ isolated, V (x, r) = µ{x} for all 0 ≤ r < r0 = r0 (x) , the smallest positive value in Λ∗ (x). Lemma 6.2 For any given x ∈ X and γ > 0, the following properties hold. (a) The function τ 7→ Rγ (x, τ ) is non-decreasing. ∗ (x) for all τ ≥ r ∗ (x). If X is compact Rγ (x, τ ) = Rγ x, rmax max If X is discrete and infinite, Rγ (x, τ ) = Rγ x, r0∗ (x) for all 0 < τ ≤ r0∗ (x).
(b) For all τ > 0, we have
Rγ (x, τ ) ≤ τ γ and, if the volume function r 7→ V (x, r) satisfies the reverse doubling property, then there exists a constant c > 0, such that Rγ (x, τ ) ≥ c τ γ
(6.2)
for all τ > 0. In the non-discrete compact case, if the volume function just satisfies the ∗ (x). In the discrete reverse doubling property at zero, (6.2) holds for all 0 < τ < rmax infinite case, if the volume function just satisfies the reverse doubling property at infinity, (6.2) holds for all τ > r0∗ (x). 27
Proof. For the first part of (a), we integrate by parts: ! Z Z V (x, s) 1 γ γ = 1− V (x, s) ds Rγ (x, τ ) = τ V (x, τ ) − dsγ , V (x, τ ) V (x, τ ) (0 , τ ] (0 , τ ] whence τ 7→ Rγ (x, τ ) is non-decreasing. The second part (a) is straightforward. Regarding (b), the general upper bound on Rγ (x, τ ) is obvious. If the volume function satisfies the reverse doubling property, then in the respective range, 1 (δτ )γ V (x, τ ) − V (x, δτ ) Rγ (x, τ ) ≥ V (x, τ ) V (x, δτ ) γ = (δτ ) 1 − ≥ δ γ (1 − κ)τ γ = c τ γ V (x, τ ) for suitable constants 0 < κ, c < 1. Now, in order to estimate the moment function t 7→ Mγ (x, t), we need to estimate a Laplacetype integral as given by the formula of Lemma 6.1. We will treat such estimates in the two technical Propositions 6.6 and 6.7 at the end of this section. Before that, in the next three theorems, we anticipate the statements of the results regarding the moment function. Theorem 6.3 Assume that (X, d) is non-compact and has no isolated points. Then the following properties hold. (1) For all x ∈ X, t > 0 and 0 < γ < 1, Mγ (x, t) ≤
tγ . 1−γ
(2) If for some x ∈ X, the volume function satisfies the reverse doubling property, then for any 0 < γ < 1, c γ Mγ (x, t) ≥ t , 1−γ for all x, t > 0 and some c > 0. Moreover, Mγ (z, t) = ∞, for all z, t > 0 and γ ≥ 1. Theorem 6.4 Assume that (X, d) is discrete and infinite. Then the following properties hold. (a) For all x, t > 0 and 0 < γ < 1, Mγ (x, t) ≤
C min {t, tγ } 1−γ
for some C > 0. (b) If for some (equivalently, all) x ∈ X the volume function satisfies the reverse doubling property at infinity, then for any 0 < γ < 1, c min {t, tγ } Mγ (z, t) ≥ 1−γ for all z , t > 0 and for some c > 0. Moreover, Mγ (z, t) = ∞ for all z, t > 0 and all γ ≥ 1. 28
Assume now that (X, d) is compact and let D be its d∗ -diameter. By Lemmas 6.1 and 6.2, for all x ∈ X, γ > 0 and t > 0, Mγ (x, t) ≤ Rγ (x, D) ≤ D γ , whence we study the behavior of the moment function t 7→ Mγ (x, t) at zero. Theorem 6.5 Assume that (X, d) is non-discrete and compact. Then the following properties hold. (1) There exists a constant C > 0 such that Ct if 1 Mγ (x, t) ≤ C t log t + 1 if C tγ if
γ > 1, γ = 1, γ < 1,
holds for all x and all 0 < t ≤ 1.
(2) If for some x ∈ X the volume function satisfies the reverse doubling property at zero, then there exists a constant c > 0 such that ct if γ > 1, 1 Mγ (z, t) ≥ c t log t + 1 if γ = 1, c tγ if γ < 1 holds for all z and all 0 < t ≤ 1.
We now provide the technical details regarding the Laplace-type estimates that imply Theorems 6.3, 6.4 and 6.5. In the following two propositions, M and R will always be two nonnegative, non-decreasing functions related by the Laplace-type integral Z ∞ t e−τ dτ . R M (t) = τ 0 Proposition 6.6 Let γ > 0 be given. (1) Assume that Asγ ≥ R(s) ,
or that respectively
R(s) ≥ B sγ
(6.3)
for some A > 0 (resp. B > 0) and all s > 0. Then the inequality A tγ ≥ M (t) , 1−γ
respectively
M (t) ≥
B tγ (1 − γ) e
holds for all 0 < γ < 1 and all t > 0. (2) Assume that there is t0 > 0 such that R(s) = 0 for all 0 < s < t0 . Assume also that one of the respective inequalities of (6.3) holds for all s > t0 . Then γ γ t t t t c′ c min , min , , respectively M (t) ≥ , M (t) ≤ 1−γ t0 t0 1−γ t0 t0 for all 0 < γ < 1, all t > 0 and some constants c, c′ > 0. 29
(3) The assumption γ ≥ 1 and the lower bound R(s) ≥ B sγ imply that M (t) = ∞ for all t > 0. Proof. It is known that for 0 < γ < 1 the Gamma-function satisfies 1 1 < Γ(1 − γ) < , (1 − γ) e 1−γ whence by monotonicity of the Laplace-type integral the first claim follows. To prove the second statement, we write Z t R e−s ds. M (t) = s {t/s≥t0 }
First assume that R(τ ) ≤ A sγ for all 0 < s < ∞ . Then we obtain γ Z Z t −s γ M (t) ≤ A e ds = A t s−γ e−s ds s {t/s≥t0 } {s≤t/t0 } −γ Z t/t0 t At0 , and ≤ A tγ s−γ ds = t 1−γ 0 0 γ Z ∞ t A tγ0 A tγ γ −γ −s = . M (t) ≤ A t s e ds ≤ 1−γ t0 1−γ 0 It follows that γ A max t0 , t−1 t t 0 M (t) ≤ min , . 1−γ t0 t0 Second, assume that R(s) ≥ B sγ , for all s ≥ t0 . Then for t/t0 ≥ 1 γ Z Z t/t0 B tγ0 t B tγ B tγ 1 −γ γ −γ −s . = s ds = M (t) ≥ B t s e ds ≥ e (1 − γ)e (1 − γ) e t0 0 0
When t/t0 ≤ 1 we get 1−γ Z Z t/t0 B tγ0 B tγ t t B tγ t/t0 −γ γ −γ −s = s ds = . M (t) ≥ B t s e ds ≥ e (1 − γ) e t0 (1 − γ) e t0 0 0
It follows that
M (t) ≥
B tγ0 min (1 − γ) e
t , t0
t t0
γ
≥
B min{t0 , 1} min (1 − γ) e
t , t0
t t0
γ
.
This proves the second claim. For the third claim observe that that if R(s) ≥ B sγ for all s ≥ t0 and γ ≥ 1, Z t/t0 γ M (t) ≥ B t s−γ e−s ds = ∞ 0
for all t > 0.
Proposition 6.7 Assume that there is t0 > 0 such that R(s) = R(t0 ) for all s ≥ t0 . Assume also that one of the respective inequalities in (6.3) holds for all 0 < s ≤ t0 . Then c1 tt0 if γ > 1, t0 c2 t log t + 1 if γ = 1, M (t) ≤ γ t if γ < 1, c 3
respectively,
M (t) ≥
c′2 t
c′3 30
t0
c′1 tt0 log tt0
+1 γ t t0
if
γ > 1,
if
γ = 1,
if
γ < 1,
for all 0 < t ≤ t0 and some positive constants c1 , c′1 , c2 , c′2 , c3 , c′3 . Proof. Let γ > 1 and 0 < t < t0 . According to our assumption Z t e−s ds + R(t0 ) 1 − e−t/t0 . R M (t) = s {t/s≤t0 }
Observe that for 0 < t < t0 ,
t t ≤ 1 − e−t/t0 ≤ . 2t0 t0 γ First, if R(s) ≤ A s for all 0 < s < t0 , then Z ∞ Z ∞ R(t0 )t R(t0 )t γ −γ −s γ M (t) ≤ A t s e ds + ≤ As s−γ ds + t0 t0 t/t0 t/t0 γ 1−γ t R(t0 )t t A t0 A tγ + = R(t0 ) + . ≤ γ − 1 t0 t0 t0 γ−1
Second, if R(s) ≥ B sγ , for all 0 < s < t0 , then M (t) ≥
R(t0 ) t . 2 t0
Assume that 0 < γ < 1 and 0 < t < t0 . Again first, if R(s) ≤ A sγ for all 0 < τ < t0 , then Z ∞ t R(t0 )t γ ≤ A tγ Γ(1 − γ) + R(t0 ) M (t) ≤ At s−γ e−s ds + t t 0 0 t/t0 γ γ γ t t A t0 t At + R(t0 ) + R(t0 ) = ≤ 1−γ t0 1 − γ t0 t0 γ ATγ t + R(t0 ) . ≤ t0 1−γ
Second, once more, when R(s) ≥ B sγ , for all 0 < s < T , then γ Z ∞ Z ∞ t B min{t0 , 1} γ −γ −s γ −γ −s M (t) ≥ B t s e ds ≥ B t s e ds ≥ . t0 e2 t/t0 1
Finally, assume that γ = 1 and 0 < t < t0 . First, if R(s) ≤ A sγ for all 0 < τ < t0 , then Z ∞ R(T )t s−1 e−s ds + M (t) ≤ At T t/T ! Z ∞ Z 1 R(t0 )t = At s−1 e−s ds + s−1 e−s ds + t0 1 t/t0 ! Z ∞ Z 1 ds R(t0 )t ds R(t0 ) t0 ≤ At + + = A+ t log + 1 . s2 t0 t0 t 1 t/t0 s
And at last, if R(s) ≥ B sγ for all 0 < τ < t0 , then Z ∞ R(t0 )t M (t) ≥ B t s−1 e−s ds + 2t0 t/t0 Z 1 ds R(t0 )t t0 R(t0 )t Bt Bt + log + = ≥ e t/t0 s 2t0 e t 2t0 R(t0 ) B Bt t0 R(to )e t0 , = ≥ min log + t log + 1 . e t 2B t0 2t0 e t
The proof is finished. Theorems 6.3, 6.4 and 6.5 follow.
31
7
The Markov generator and its spectrum
It is known that any strongly continuous semigroup {Pt }t>0 in a Banach space B has the infinitesimal generator (−L, DomL ), whose domain is f − Pt f exists strongly , DomL = f ∈ B : lim t→0 t and, for f ∈ DomL , f − Pt f . t In general, (L, DomL ) is an unbounded densely defined closed operator in B. Moreover, if B is a Hilbert space and the operators Pt are symmetric then (L, DomL ) is a self-adjoint operator and Pt = exp(−tL). The purpose of this section is to study the infinitesimal generator (−L, DomL ) of our symmetric Markov semigroup {P t }t>0 given by (1.4). We will refer to (L, DomL ) as the Laplace operator (shortly, Laplacian) associated to the symmetric Markov semigroup {P t }t>0 . This is a non-negative definite self-adjoint operator. Let us introduce the function Z 1/d∗ (x,y) N (x, τ ) exp(−τ t) dτ , Jt (x, y) := Lf = lim
t→0
0
defined for all x, y ∈ X and all t ≥ 0. In particular, for t = 0 we set Z 1/d∗ (x,y) N (x, τ ) dτ . J(x, y) := J0 (x, y) =
(7.1)
0
By construction, and in particular by (2.4), the functions Jt (x, y) and J(x, y) are symmetric and finite on the set {(x, y) ∈ X 2 : x 6= y}, and J(x, x) = ∞ for all x, whereas Jt (x, x) may be finite for some t > 0 and x ∈ X. We shall see that J(x, y) plays the role of the jump-kernel of the Dirichlet form associated with our Laplacian. Yet one more observation is in order: for all x, y, λ > 0, J(x, y) > λ2 Gλ (x, y), and in the non-compact case, for any fixed λ > 0, J(x, y) ∼ λ2 Gλ (x, y)
as d∗ (x, y) → ∞
by Theorem 5.1. Lemma 7.1 The following properties hold. (a) For all x, y ∈ X, 1 1 1 1 N y, N y, ≤ J(x, y) ≤ . 2d∗ (x, y) 2d∗ (x, y) d∗ (x, y) d∗ (x, y) (b) Let B = Br∗ (a) be a fixed ball of radius r > 0. Then Z 1 J(x, y) dµ(y) ≤ for any x ∈ B, and r X\B 2 Z Z V (a, r) . J(x, y) dµ(y) dµ(x) ≤ r2 B X\B 32
Proof. Statement (a) is immediate from the definition of the function J(x, y) and the fact that the function τ → N (x, τ ) is non-decreasing. For (b), we may of course assume that B ( X. Recall the structure of the sets Λ(a) and Λ∗ (a) from (2.2) and (3.1). Let once more B ′ be the predecessor ball of B and r ′ its radius (see Definition 2.1). It exists because B ( X. All x ∈ B and y ∈ X \ B satisfy d∗ (x, y) = d∗ (a, y) ≥ r ′ > r, whence J(x, y) = J(a, y). With this observation in mind we compute for any z ∈ B ′ \ B Z
Z
Z
1/s
dτ (7.2) V (a, 1/τ ) 0 ,∞) Z 1/r′ Z Z 1/r′ dV (a, s) V (a, 1/τ ) − V (a, r ′ ) dτ = = dτ V (a, 1/τ ) (r ′ , 1/τ ] V (a, 1/τ ) 0 0 Z 1/r′ 1 dτ 1 ′ = − V (a, r ) = ′ − V (a, r ′ )J(a, z), ′ r V (a, 1/τ ) r 0
J(x, y) dµ(y) =
dV (a, s)
(r ′
X\B
Since r ′ > r, this proves the first identity of (b). To prove the second identity of (b), we observe that when y ∈ B and x ∈ X \ B, 1 1 N a, . J(x, y) = J(x, a) ≤ d∗ (x, a) d∗ (x, a) Using the inequality 7.3 we write 2 Z Z Z J(x, y)dµ(y) dµ(x) ≤ V (a, r)2 X\B
B
≤
V (a, r) r
X\B 2 Z
1 1 N a, d∗ (x, a) d∗ (x, a)
X\B
2
(7.3)
dµ(x)
dµ(x)
2 . V a, d∗ (x, a)
Write Λ∗ (a) = {0} ∪ {rk }, where k varies as in (2.2), but refers to the intrinsic metric d∗ . Then Z X V (a, rk ) − V (a, rk−1 ) dµ(x) 2 2 = X\B V a, d∗ (x, a) V (a, rk ) rk ≥r ′ X V (a, rk ) − V (a, rk−1 ) 1 < = , V (a, r )V (a, r ) V (a, r) k k−1 r >r k
which concludes the proof. As a set of test functions, we introduce the linear space D of all real valued, locally constant and compactly supported functions on X. Evidently D is dense in all Banach spaces Lp = Lp (X, µ), 1 ≤ p < ∞, as well as in C0 (X), the space of all continuous functions vanishing at infinity. The following theorem shows that the Laplacian arises from a difference operator, and it leads to an explicit representation of the associated Dirichlet form. Theorem 7.2 Let (L, DomL ) be the Laplacian. Then D ⊂ DomL and for any f ∈ D , Z f (x) − f (y) J(x, y) dµ(y), and Lf (x) = XZ Z 2 1 f (x) − f (y) J(x, y) dµ(x) dµ(y). (Lf, f ) = 2 X×X 33
Proof. For any continuous compactly supported function f we write Z p(t, x, y) 1 t f (x) − f (y) dµ(y). f (x) − P f (x) = t t X
Using the equation (4.1) we obtain 1 f (x) − P t f (x) = t =
Z
f (x) − f (y)
ZX
Z
1/d∗ (x,y)
N (x, τ ) exp(−τ t) dλ dµ(y)
0
f (x) − f (y) Jt (x, y) dµ(y).
X
Now assume that f is a test function. Then f can be written as a finite linear combination of indicators 1Bk of non-intersecting balls Bk . Hence we can restrict our considerations to the case when f = 1B , where B = Br∗ (a) is a ball of radius r > 0. Under this assumption the function Z f (x) − f (y) Jt (x, y) dµ(y) Ut (x) = X
has the form
Ut (x) =
( R
X\B
−
R
B
Jt (x, y) dµ(y)
if
x∈B
Jt (x, y) dµ(y)
if
x ∈ X \ B.
As t → 0 the function Ut converges pointwise to the similar function U , ( R if x∈B X\B J(x, y) dµ(y) U (x) = R − B J(x, y) dµ(y) if x ∈ X \ B.
(7.4)
By construction, Ut ∈ L2 for all t > 0, and U ∈ L2 by Lemma 7.1. Our above computations show that 2 Z Z
U − Ut 2 2 J(x, y) − Jt (x, y) dµ(y) dµ(x) = L (X,µ) B
+
Z
X\B
X\B
Z
B
2 J(x, y) − Jt (x, y) dµ(y) dµ(x).
Since J(x, y) > Jt (x, y) and Jt (x, y) → J(x, y) monotonically as t → 0, we can apply the Monotone Convergence Theorem to conclude that Ut → U in L2 . Thus L1B (x) = U (x)
(7.5)
and we see that indeed D ⊂ DomL and that the first formula holds for every test function. We now prove the second formula. It is a general fact concerning any symmetric Markov operator which admits a transition density with respect to a reference measure that, in our specific setting, the following identity holds for any test function f : ZZ 2 1 t f − P f, f = f (x) − f (y) p(t, x, y) dµ(x) dµ(y) , whence 2 Z ZX×X 2 1 1 f − P t f, f = f (x) − f (y) Jt (x, y) dµ(x) dµ(y). t 2 X×X We apply once more the Monotone Convergence Theorem and the fact that to get the desired result. 34
1 t (f
− P t f ) → Lf
Let B be the family of all closed balls in (X, d), or equivalently, in (X, d∗ ). For any B ∈ B we let r ∗ (B) denote the d∗ -diameter of B, or equivalently, its minimal radius, a number that belongs to the set Λ∗ of (3.1). For any B ∈ B, with B ( X in the compact case, we define the function 1 1 (7.6) 1B − 1B ′ , fB = µ(B) µ(B ′ ) where the ball B ′ is the the predecessor of B. By abuse of notation, we shall write X ′ = X. When X is compact, it also belongs to B, and we set 1 1X . µ(X)
fX =
Observe that fB ∈ D, its support is B ′ and, unless B = X, kfB k2L2 =
1 1 − , µ(B) µ(B ′ )
while of course kfX k2L2 = 1/µ(X) and LfX = 0 in the compact case. For B, C ∈ B, we have fB ⊥ fC if and only if B ′ 6= C ′ . Theorem 7.3 (1) The family of functions {fB : B ∈ B} is a complete system in L2 (i.e., its linear span is dense). (2) Any function fB , B 6= X, satisfies the equation LfB =
1
fB . r ∗ (B ′ )
In particular, the symmetric operator (L, D) has a complete system of eigenfunctions. Proof. (1) Assume that some function f ∈ L2 is orthogonal to all functions fB . Then for any B ∈ B, B 6= X, the averages Z Z 1 1 f dµ and f dµ µ(B) B µ(B ′ ) B ′ coincide. Since any two balls are contained in a common bigger ball, by induction the averages coincide over any two balls in B. It follows that f must be a constant function. X is non-compact, then f = 0 because it belongs to L2 (X, µ) and µ(X) = ∞ by assumption (2.1). If X is compact, then f = 0 because it is orthogonal to the function fX = 1. This proves (1). (2) For a given ball B 6= X, LfB (x) =
1 1 L1B (x) − L1B ′ (x). µ(B) µ(B ′ )
We distinguish the following three cases regarding the location of x. Case1. x ∈ X \ B ′ . Then fB (x) = 0. By (7.4) and (7.5), Z Z 1 1 J(x, y) dµ(y). J(x, y) dµ(y) + LfB (x) = − µ(B) B µ(B ′ ) B ′ 35
Since J(x, y) = J(x, a), for any y, a ∈ B, LfB (x) = −
µ(B ′ ) µ(B) J(x, a) + J(x, a) = 0. µ(B) µ(B ′ )
1 1 Case2. x ∈ B. Then fB (x) = µ(B) − µ(B ′) . Again by (7.4) and (7.5), Z Z 1 1 LfB (x) = J(x, y) dµ(y) − J(x, y) dµ(y) µ(B) X\B µ(B ′ ) X\B ′ Z Z 1 1 1 − J(x, y) dµ(y) + J(x, y) dµ(y). = µ(B) µ(B ′ ) µ(B) B ′ \B X\B ′
When x ∈ B and y ∈ X \ B ′ , we have d∗ (x, y) = d∗ (a, y) for any a ∈ B. Thus, by (7.2) , for any z ∈ B ′ \ B, 1 1 1 1 ′ ′ − − µ(B )J(a, z) + µ(B ) − µ(B) J(a, z) LfB (x) = µ(B) µ(B ′ ) r ∗ (B ′ ) µ(B) 1 1 1 1 = ∗ ′ − = ∗ ′ fB (x). ′ r (B ) µ(B) µ(B ) r (B ) 1 Case 3. x ∈ B ′ \ B. Then fB (x) = − µ(B ′ ) . Once more by (7.4) and (7.5),
1 LfB (x) = − µ(B)
Z
1 J(x, y) dµ(y) − ′) µ(B B
Z
J(x, y) dµ(y). X\B ′
Our x belongs to a ball C 6= B with C ′ = B ′ (there is a finite number of such balls). We apply (7.2) and choosing an arbitrary z ∈ B ′ \ C, we obtain 1 1 ′ − µ(B )J(x, z) LfB (x) = −J(x, z) − µ(B ′ ) r ∗ (B ′ ) 1 1 1 = − = ∗ ′ fB (x). r ∗ (B ′ ) µ(B ′ ) r (B ) This completes the proof of statement (2). For B ∈ B, we define the space HB = Span{fC : C ∈ B , C ′ = B}.
(7.7)
An easy computation shows that X
µ(C)fC = 0.
(7.8)
C∈B : C ′ =B ′ = B, then by another straightforward On the other hand, if we select one CB ∈ B with CB computation the functions fC : C ′ = B , C 6= CB are linearly independent. Combining these observations with Theorem 7.3, we now see that the spectrum of our Laplacian is pure point.
36
Corollary 7.4 (a) The symmetric operator (L, D) is essentially self-adjoint. (b) The spectrum of the self-adjoint operator (L, DomL ) coincides with the closure of the set 1 ∗ ∗ : r ∈ Λ∗ , r 6= 0 ∪ {0} with Λ∗ as in (3.1). r∗ (c) The space L2 (X, µ) decomposes as a direct sum of finite-dimensional eigenspaces which are spanned by compactly supported functions: M L2 (X, µ) = HB . B∈B
It is now easy to prove our general subordination result. Recall the inverse exponential distribution σ ∗ of (1.5). Also recall from the introduction that our (d, µ, σ)-process and the associated Laplacian coincides with the (d∗ , µ, σ ∗ )-process, which is the standard (d∗ , µ)-process in our terminology. Theorem 7.5 Let ψ : [0 , ∞) → [0 , ∞) be an increasing bijection, and let (L, DomL ) be the Laplacian associated with the (d, µ, σ)-process. Then the self-adjoint operator ψ(L), Domψ(L ) is the Laplacian associated with the standard (dψ , µ)-process, where the new ultra-metric dψ on X is given by 1 1 =ψ . dψ (x, y) d∗ (x, y) (It is the intrinsic metric of the new process.) Proof. We apply the L2 -decomposition of Corollary 7.4(c) and the fact that each space HB ⊂ L2 is an eigenspace of the operator (L, D) corresponding to the eigenvalue 1/r ∗ (B ′ ). Let ΠB : L2 → HB be the corresponding orthoprojector. Then, for any function f ∈ DomL , X X 1 ΠB f. (7.9) f= ΠB ′ f and Lf = r ∗ (B ′ ) B∈B
B∈B
(The sums are at most countable by the general theory.) Let rψ (B ′ ) be the diameter of the ball B ′ with respect to the new ultra-metric dψ . By Spectral Theory, for any f ∈ Domψ(L) , X X 1 1 ΠB f. (7.10) ΠB f = ψ(L)f = ψ ∗ ′ r (B ) dψ (B ′ ) B∈B
B∈B
Comparing the equations (7.9) and (7.10), we now conclude that the self-adjoint operator ψ(L), Domψ(L) coincides with the Laplacian of the standard (dψ , µ)-process on X.
Remark 7.6 Recall that a non-negative definite, self-adjoint operator L is a Laplacian if its semigroup (e−tL )t>0 is Markovian. In general, by Bochner’s theorem, for any Laplace operator L the operator ψ(L) is again a Laplace operator, provided that ψ is a Bernstein function. See ˇek [50]. Now, ψ(λ) = λα is a Bernstein function for example Schilling, Song and Vondrac if and only if 0 < α ≤ 1. Thus, for a general Laplacian, it is not true that Lα with α > 1 is a Laplace operator. However, by Theorem 7.5, in our specific case of the isotropic semigroup {P t }t>0 plus associated Laplacian on an ultra-metric space, as given by (1.4), the operator Lα is a Laplace operator for any α > 0. 37
Proposition 7.7 Let (X, d) be a non-compact, proper ultra-metric space. Let M ⊆ [0, ∞) be any closed set (unbounded, if X contains at least one non-isolated point) that accumulates at 0. Then the following holds. (a) There exists a proper ultra-metric d′ on X which generates the same topology as d and a Laplacian (L, DomL ) on (X, d′ ) such that Spec(L, DomL ) = M. (b) Suppose in addition that there exists a partition of X into d-balls that consists of infinitely many non-singletons. Then the ultra-metric d′ of (a) can be chosen such that the collections of d-balls and d′ -balls coincide. Proof. The set D = {x ∈ (0, ∞) : x−1 ∈ M } ∪ {0} is a closed, unbounded subset of [0 , ∞) containing 0. What we need for (a) is that there exists a proper ultra-metric d′ on X that generates the same topology as d and such that the closure of the value set Λd′ of that metric coincides with D. This metric property is proved by Bendikov and Krupski [8, §2]. Given µ, the Laplacian associated with the standard (d′ , µ)-processes has the required property by Corollary 7.4. A proof of the additional statement (b) on the ultra-metric can also be found in [8, §2]. It is known that any continuous symmetric Markov semigroup can be extended to all spaces 1 ≤ p < ∞, as a continuous contraction semigroup. In particular, this is true for the semigroup (P t ). We use the same notation for the extended semigroup and denote by (−Lp , DomLp ) its infinitesimal generator. Lp ,
Theorem 7.8 For any 1 ≤ p < ∞, Spec(Lp , DomLp ) = Spec(L2 , DomL2 ). Proof. Let ΠB : L2 → HB be the orthoprojector onto the space defined in (7.7). Consider the following family of orthoprojectors X Es = ΠB . B∈B : r ∗ (B)>1/s
An immediate consequence of (7.9) is that for any f ∈ DomL2 , Z ∞ s d(Es f, f ). (L2 f, f ) =
(7.11)
0
Claim. For any f ∈ L2 , Es f = Q1/s f.
(7.12)
Indeed, the linear subspace Q1/s (L2 ) ⊂ L2 consists of all functions in L2 each of which is constant on any ball B of radius 1/s. On the other hand, the linear subspace Es (L2 ) ⊂ L2 can be represented as X X Es (L2 ) = ΠB (L2 ) = HB , B∈B : r ∗ (B)>1/s
38
r ∗ (B)>1/s
It follows that Es (L2 ) is spanned by the set {fB : r ∗ (B) > 1/s}. All those functions take constant values on any ball B of radius 1/s. Thus Es (L2 ) ⊂ Q1/s (L2 ). For any ball B of radius r ∗ (B) > 1/s we can write X 1B = µ(B) fC . C∈B : B⊆C
It follows that Q1/s (D) ⊂ Es (L2 ), whence Q1/s (L2 ) ⊂ Es (L2 ) as well. Thus Es (L2 ) = Q1/s (L2 ). Since both Es and Q1/s are orthoprojectors, they coincide. This proves the claim. Since each eigenfunction fB is compactly supported, Σ := Spec(L2 , DomL2 ) ⊂ Spec(Lp , DomLp ). Let λ ∈ / Σ, so that δ = min{|λ − s| : s ∈ Σ} > 0. Let f and g be bounded functions supported by some ball B = Bρ (a), where a ∈ X and ρ ∈ Λ∗ (a). Using (7.11), we can write Z d(Es f, g) (L2 − λ · id)−1 f, g = s−λ [0 , ∞) ! Z Z d(Es f, g) + = = I1 + I2 . s−λ [1/ρ , ∞) [0 , 1/ρ) (I) Estimate of I1 . For s < 1/ρ we have evidently B = Bρ (a) ⊂ B1/s (a), whence (Es f, g) =
1 (f, 1)(g, 1) = N (a, s) (f, 1)(g, 1). V (a, 1/s)
It follows that I1 = (f, 1)(g, 1)
Z
[0 , 1/ρ)
1 dN (a, s). s−λ
We use this equality for conjugate exponents (p, q) and obtain 1
1
|I1 | ≤ kf kLp V (a, ρ) p kgkLq V (a, ρ) q
δ −1 = δ −1 kf kLp kgkLq . V (a, 1/τ )
(II) Estimate of I2 . We observe that by (7.12)), the signed measure dE(s) = 1[1/r , ∞) d(Es f, g) is concentrated on the set Mρ =
1 : r ∈ Λ∗ (a) , r ≤ ρ . r
Once more, write Λ∗ (a) = {0}∪{rk }, where k varies as in (2.2), but the increasing sequence {rk } refers to d∗ . By assumption, there is i such that ρ = ri , and if we re-parametrize sk = 1/ri−k then 1 : k ≤ i = {sk : k ≥ 0} . Mρ = rk When X is discrete, Mρ is finite. Otherwise, sk → ∞ as k → ∞. Having these observations in mind, we write 39
I2 =
X k≥0
1 Esk+1 f, g − Esk f, g . sk − λ
Next we apply the Abel transformation with s−1 = 0 and obtain X 1 1 1 I2 = − Esk f, g . E1/ρ f, g + λ sk−1 − λ sk − λ k≥0
Since λ ∈ / Σ, we do not have λ = sk for any k. Thus, there is k0 ≥ 0 such that sk0 −1 < λ < sk0 ,
and we write I2
X 1 1 1 − E1/ρ f, g + Esk f, g + = λ sk−1 − λ sk − λ k≤k0 X 1 1 + − Esk f, g . sk−1 − λ sk − λ k>k0
By (7.12), Es = PB1/λ is a Markov operator, whence
|(Es f, g)| ≤ kf kLp kgkLq . We note that for any k ≥ k0 + 1 we have sk−11 −λ − sk1−λ > 0. After all these preparations we can estimate I2 : X 1 1 1 −1 (Es f, g) |I2 | ≤ + 2(k0 + 1)δ − kf kLp kgkLq + k λ sk−1 − λ sk − λ k>k0 1 1 + 2(k0 + 1)δ −1 + kf kLp kgkLq ≤ λ s k0 − λ 1 −1 + (2k0 + 3)δ kf kLp kgkLq . ≤ λ Putting together the bounds obtained in (I) and (II), we finally come to the desired inequality 1 −1 −1 + (2k0 + 4)δ kf kLp kgkLq . (L2 − λ id) f, g ≤ λ
This inequality shows that the operator (L2 − λ id)−1 can be extended to Lp as a bounded operator. The extended operator evidently coincides with (Lp − λ id)−1 . Thus λ ∈ / Spec(Lp , DomLp ) and therefore Spec(Lp , DomLp ) ⊆ Spec(L2 , DomL2 ). The proof of the theorem is finished. We remark that (7.11) and (7.12) indeed provide explicitly the spectral resolution, as outlined in the Introduction. The last theorem of this section concerns harmonic functions. Notice that the semigroup t (P )t>0 admits an extension to L∞ as a contraction semigroup, but this extension is not a continuous semigroup unless X is discrete. We define the Laplacian (L∞ , DomL∞ ) as a weak infinitesimal generator, that is, we define DomL∞ ⊂ L∞ as the set of functions f such that f (x) − P t f (x) t→0 t ∞ exists for all x ∈ X and belongs to L . This defines L∞ f (x) for f ∈ DomL∞ and x ∈ X. L∞ f (x) = lim
40
Theorem 7.9 (Strong Liouville property) Any measurable function f : X → [0 , ∞] which satisfies P f = f must be constant. In particular, 0 ∈ Spec(L∞ , DomL∞ ) is an eigenvalue of multiplicity 1. Proof. We may assume without loss of generality that we are dealing with the standard (d∗ , µ)process on X, where d = d∗ is the intrinsic metric and the distance distribution is σ ∗ , the inverse exponential distribution (1.5) . Step 1. We first prove the claim assuming that f takes only finite values. By assumption, Z ∞ Qs f dσ ∗ (s), f=
(7.13)
0
see (1.3). For any fixed r > 0 we apply to both sides of this equation the operator Qr . Using (2.6), we obtain Z ∞ Z ∞ Qs f dσ ∗ (s), Qmax{r,s} f dσ ∗ (s) = σ ∗ (r)Qr f + Qr f = r
0
whence,
Z
1 − σ ∗ (r) Qr f =
∞
Qs f dσ ∗ (s).
(7.14)
r
Let B ∈ B be a ball with diameter r, and let r ′ be the diameter of the predecessor ball B ′ . We consider (7.14) for r and for r ′ , and take the difference: for any x ∈ B, Z r′ ′ 1 − σ ∗ (r) Qr f (x) − 1 − σ ∗ (r ) Qr′ f (x) = Qs f (x)dσ ∗ (s) = σ ∗ (r ′ ) − σ ∗ (r) Qr f (x), r
or equivalently
We conclude that
1 − σ ∗ (r ′ ) Qr f (x) − Qr′ f (x) = 0. Qr f (x) = Qr′ f (x)
(7.15)
for every x ∈ B and all r, r ′ ∈ Λ∗ (x), where initially r ′ is the smallest element in Λ∗ (x) with r ′ > r. Inductively, we get that (7.15) holds for all r, r ′ ∈ Λ∗ (x) without further restriction, that is, the function r 7→ Qr f (x) is constant. Since x 7→ Qr f (x) is constant on each ball of radius r, we get that the latter function is constant both in x and in r. Since f (x) = P f (x), f is constant as well. Step 2. Assume now that f (a) < ∞ for some a ∈ X. Let B = Br (a) and B ′ = Br′ (a) be as above in Step 1. By (7.13), Z r′ Z ∞ ′ Qs f (a) dσ ∗ (s) = f (a) < ∞ . σ ∗ (r ) − σ ∗ (r) Pr f (a) Qs f (a) dσ ∗ (s) ≤ 0
r
Thus, Qr f (a) < ∞. Recalling the structure (2.2) of Λ∗ (a), again by induction, Qr f (a) < ∞ for any r > 0. In particular, for any r > 0, the function fr (x) = Qr f (x) takes only finite values and evidently satisfies the equation P fr = fr . By the first part of the proof, fr is constant. The identity (7.13) now yields that f is constant. For the last statement of the theorem, assume that f ∈ DomL∞ and L∞ f = 0. Since f is a bounded function we can assume that it is non-negative. Since f ∈ DomL∞ we can write Z 1 Z 1 Qs L∞ f ds = 0, L∞ Qs f ds = − Pf − f = − 0
0
whence, by Step 1, f is constant. This concludes the proof. 41
Corollary 7.10 Assume that for some measure µ′ the semigroup {P t }t>0 is symmetric in L2 (µ′ ). Then µ′ is proportional to µ.
8
The p-adic fractional derivative
Consider the field Qp of p-adic numbers endowed with the p-adic norm kxkp and the p-adic ultra-metric d(x, y) = dp (x, y) = kx − ykp . Let µp be the Haar measure on Qp , normalized such that µp (Zp ) = 1. Let D be the set of test functions on Qp , that is, locally constant functions with compact support. The notion of p-adic fractional derivative, closely related to the concept of p-adic Quantum Mechanics, was introduced in several mathematical papers by Vladimirov [54], Vladimirov and Volovich [55] and Vladimirov, Volovich and Zelenov [56]. In particular, in [54] a one-parametric family {(Dα , D)}α>0 of operators, called operators of fractional derivative of order α, has been introduced. Definition 8.1 The operator (Dα , D), α > 0, is defined via the Fourier transform on the locally compact Abelian group Qp by α f (ξ) = kξkα fb(ξ), ξ ∈ Q . d D p p
It was shown by the above named authors that each operator (Dα , D) can be written as a Riemann-Liouville type singular integral operator Z f (x) − f (y) pα − 1 α (8.1) D f (x) = 1+α dµp (y). −α−1 1−p Qp kx − ykp The aim of this section is in particular to show that the operator (Dα , D) is in fact the restriction to D of the Laplacian (i.e., −Dα is the Markov generator) of an appropriate isotropic Markov semigroup {Pαt }t>0 , as constructed and studied throughout this paper. We use the following distance distribution function α exp − pr if r > 0 , σ α (r) = 0 if r ≤ 0 . We now consider the isotropic Markov semigroup {Pαt }t>0 , associated with the triple (dp , µp , σ α ) as constructed in (1.2) – (1.4), that is, Z ∞ Qr f (x) dσ tα (r). Pαt f (x) = 0
The semigroup acts in L2 , and we let (Lα , DomLα ) be the corresponding Laplacian. Theorem 8.2 For any α > 0, (Lα , D) = (Dα , D). Proof. By Theorem 7.2, for f ∈ D, Lα f (x) =
Z
Qp
f (x) − f (y) Jα (x, y) dµp (y), 42
so that we are left to compute the function Z Z 1/d∗ (x,y) N (x, τ ) dτ = Jα (x, y) =
1/d∗ (x,y)
0
0
dτ , ∗ (x) µp B1/τ
∗ (x) is the d -ball at x of radius 1/τ . By Example (4.11), where B1/τ ∗
d∗ (x, y) =
kx − ykp p
α
,
whence ∗ B1/τ (x) = Bp/τ 1/α (x),
where Bp/τ 1/α (x) is the dp -ball at x of radius p/τ 1/α . Putting all these facts together, we obtain Jα (x, y) =
Z (p/kx−yk )α p 0
dτ
=p µp Bp/τ 1/α (x)
α
Z
∞ kx−ykp
αr −α−1 dr . µp Br (x)
The value set of the metric dp (x, ·) is Λdp (x) = {pk : k ∈ Z} for every x ∈ Qp . Thus, if kx − ykp = pk then the last integral is Z
∞ pk
αr −α−1 dr µp Br (x)
=
XZ
n≥k
pn+1
pn
αr −α−1 dr µp Br (x)
1 1 αr −α−1 dr X 1 = − pn pn pnα p(n+1)α n n≥k n≥k p −k(α+1) 1 1 p 1 X = 1− α = 1− α n(α+1) p p p 1 − p−(α+1) n≥k !α+1 α+1 1 − p−α 1 − p−α 1 1 = = . 1 − p−(α+1) pk 1 − p−(α+1) kx − ykp =
XZ
pn+1
Thus finally we obtain the following equality pα − 1 Jα (x, y) = 1 − p−α−1
1 kx − ykp
!α+1
.
(8.2)
In view of (8.1), this proves the claim. As a consequence of Theorem 8.2, Corollary 4.5 and Example 4.11 we obtain the following heat kernel estimate. Theorem 8.3 The semigroup {Pαt }t>0 admits a continuous transition density pα (t, x, y) with respect to Haar measure µp which satisfies (t1/α
c2 t c1 t ≤ pα (t, x, y) ≤ 1/α , 1+α + kx − ykp ) (t + kx − ykp )1+α
(8.3)
for some constants c1 , c2 > 0. The upper bound in (8.3) was also obtained by a different method by Kochubey [37, Ch.4.1, Lemma 4.1]. 43
Theorem 8.4 The semigroup {Pαt }t>0 is transient if and only if 0 < α < 1. In the transient case, its Green function is given explicitly as 1 − p−α Gα (x, y) = 1 − pα−1
1 kx − ykp
!−α+1
.
(8.4)
For an extension of Lα acting in the space of distributions D ′ , the formula (8.4) for a fundamental solution of Lα was obtained by Vladimirov [54, Thm 1, p.51] and Kochubey [37, Ch.2.2]. Proof. The characterization of transience follows from Example 5.7. In this case, by Theorem 5.4(3) α
G (x, y) =
Z
0
=
Z
Z ∞ 1 N (x, τ ) dτ N x, = dτ τ2 τ d∗ (x,y) Z ∞ dτ dτ . = α V (x, τ ) µ B (kx−ykp /p) p pτ 1/α (x)
1/d∗ (x,y)
∞
d∗ (x,y)
With the change of variables s = pτ 1/α , if kx − ykp = pk for k ∈ Z, we obtain α
G (x, y) = =
=
Z
Z n+1 α−1 αsα−1 ds 1 X p αs ds = α n p p n pk µp Bs (x n≥k p 1−α pα − 1 1 1 X 1 (n+1)α nα = p − p pα pn pα (1 − pα−1 ) pk n≥k !−α+1 1 − p−α 1 . 1 − pα−1 kx − ykp 1 pα
∞
The proof is finished. Let {Xt } be the Markov process on Qp driven by the Markov semigroup {Pαt }t>0 . The semigroup is translation invariant, whence the process has independent and stationary increments. For any given γ > 0 and t > 0 , consider the moment of order γ of Xt defined in terms of the p-adic distance dp (x, y): Mγ (t) = E(kXt kγp ), where as in the Section 6, E is expectation with respect to the probability measure on the trajectory space of the process starting at 0. Applying Theorem 6.3 we obtain the following estimates. Theorem 8.5 The moment Mγ (t) is finite if and only if γ < α. Moreover, if γ < α, there exists a constant κ = κ(α) > 0 such that κtγ/α αtγ/α ≤ Mγ (t) ≤ . α−γ α−γ Let us write Dαq for the operator where −Dαq is the infinitesimal generator of the semigroup {Pαt }t>0 acting in Lq (µp ), 1 ≤ q < ∞. Applying Theorems 7.3 and 7.8) we obtain the following spectral result.
44
Theorem 8.6 For any α > 0 and 1 ≤ q < ∞, Spec(Dαq ) = {pkα : k ∈ Z} ∪ {0}. Each λk = pkα is an eigenvalue with infinite multiplicity. In the general setting of Theorems 7.3 and 7.8, some eigenvalues may well have finite multiplicity and some not. Indeed, attached to each ball B of d∗ -diameter 1/λ there are the eigenvalue λ and the corresponding finite dimensional eigenspace HB . This eigenspace is spanned by the finitely many functions 1 1 1C − 1B , fC = µ(C) µ(B) where C runs through all balls whose predecessor (in the sense of Definition 2.1) is C ′ = B. Recall that dim HB = l(B) − 1, where l(B) = ♯{C ∈ B : C ′ = B}. It follows that in general, if there exists only a finite number of distinct balls of d∗ -diameter 1/λ then the eigenvalue λ has finite multiplicity. This is clearly not the case for the ultra-metric measure space (Qp , dp , µp ) and the semigroup t {Pα }t>0 . Indeed, every d∗ -ball has its diameter in the set Λα = {pkα : k ∈ Z}, and each ball Br (0) has infinitely many disjoint translates {ai + Br (0) : i = 1, d, . . . }, which cover Qp and are balls of the same diameter. Thus, all eigenvalues have infinite multiplicity. Remark 8.7 Let H(λ) be the eigenspace corresponding to the eigenvalue λ ∈ Λα . Then 2
L =
M
H(λ)
and H(λ) =
∞ M
Hai +B1/λ (0) .
(8.5)
i=1
λ∈Λα
As after (7.8), we choose for each closed ball B ⊂ Qp an orthonormal basis {eB i : 1 ≤ i ≤ p − 1} in HB . In view of (8.5), the set of eigenfunctions {eB : B ∈ B, 1 ≤ i ≤ p − 1} is an orthonormal i 2 basis in L . (This reasoning applies to arbitrary ultra-metric spaces.) Whether this set is a Schauder basis in Lq , 1 ≤ q < ∞, is an open question. Rotation invariant Markov semigroups. Let {Pt }t≥0 be a symmetric, translation invariant Markov semigroup on the additive Abelian group Qp . This semigroup acts in C0 (Qp ), the Banach space of continuous functions vanishing at ∞. It follows that there exists a weakly continuous convolution semigroup {pt }t>0 of symmetric probability measures on Qp such that Pt f (x) = pt ∗ f (x).
(8.6)
As the probability measures pt are symmetric, the following identity holds, which is basic in the theory of infinite divisible distributions: pbt (ζ) = exp −t Ψ(ζ) ,
where Ψ : Qp 7→ R+ is a non-negative definite symmetric function on Qp . By the L´evy-Khinchin formula, Z 1 − Rehx, ζi dJ(x), Ψ(ζ) = Qp \{0}
where J is a symmetric Radon measure on Qp \ {0} – the Levy measure associated with the non-negative definite function Ψ. See the book of Berg and Forst [12]. 45
Definition 8.8 For any a ∈ Qp with kakp = 1 define the rotation operator θa : Qp → Qp by θa (x) = ax. We say that the Markov semigroup {Pt } as above is rotation invariant if θ a (pt ) = pt for all a ∈ Qp with kakp = 1,
(8.7)
Let L be the (positive definite) generator of Pt , that is, Pt = exp (−tL). It is easy to see that (8.7) is equivalent to θa ◦ L = L ◦ θ a . In this case we also say that L is rotation invariant. By construction, any isotropic Markov semigroup P t defined on the ultra-metric measure space (Qp , dp , µp ) is rotation invariant. As we will see the class of all isotropic Markov semigroups is indeed a proper subset of the class of rotation invariant Markov semigroups. Assume that the semigroup {Pt } is rotation invariant. Then for all a such that kakp = 1 we will have Ψ(aζ) = Ψ(ζ) and θ a (J) = J. (8.8) Since the Haar measure µp of each sphere is strictly positive, (8.7) and (8.8) imply that the measures pt and J are absolutely continuous with respect to µp and have densities pt (x) and J(x) which depend only on kxkp . The same is true for the function Ψ, so that J(x) = j(kxkp ) and
Ψ(ζ) = ψ(kζkp ).
Let (L, DomL ) be the Laplacian associated with the semigroup {Pt }. All the above shows that D ⊂ DomL and Lu = ψ(D)u, u ∈ D, (8.9) where D = D1 is the operator of fractional derivative of order α = 1. By (8.9), the operator (L, D) has a complete system of eigenfunctions {fC : C ∈ B}. Associated with each ball B of radius pm , there is the (p − 1)-dimensional eigenspace HB spanned by all functions fC , where C runs through all balls whose predecessor (in the sense of Definition 2.1) is C ′ = B, and the corresponding eigenvalue is λ(m) = ψ(p−m+1 ). Let {a(m)}m∈Z be a sequence of real numbers satisfying a(m) ≥ a(m + 1),
a(+∞) = 0
and
0 < a(−∞) = W ≤ +∞.
(8.10)
Define the sequence {λ(m)}m∈Z by λ(m) = a(m) − (p − 1)−1 {a(m + 1) − a(m)}.
(8.11)
Theorem 8.9 In the above notation, when {a(m)} varies among all sequences satisfying (8.10), the sequence {λ(m)} of (8.11) provides a complete description of the set Spec(L) of the class of all rotation invariant Laplacians L (resp. all rotation invariant Markov semigroups Pt = exp(−tL)). Proof. Consider a rotation invariant Laplacian L = ψ(D). We compute the non-negative definite function Ψ(ζ) = ψ(kζkp ) associated with L. We have Z 1 − Rehx, ζi j(kxkp ) dµp (x) ψ(kζkp ) = Qp \{0} Z X = 1 − Rehx, ζi dµp (x). j(pk ) k∈Z
{x:kxkp =pk }
46
According to Vladimirov [54], Example 4, k k−1 if kζkp ≤ p−k , p −p −pk−1 if kζkp = p−k+1 , hx, ζi dµp (x) = {x:kxkp =pk } 0 if kζkp ≥ p−k+2 .
Z
In particular, we have of course
Z
{x:kxkp
=pk }
dµp (x) = pk − pk−1 .
Let kζkp = p−m+1 , then the above computations yield ψ(p−m+1 ) = j(pm ) pm + 1 − p−1
X
j(pk ) pk .
(8.12)
k≥m+1
Define the non-increasing sequence {a(m)}m∈Z by a(m) = 1 − p−1
X
j(pk ) pk = 1 − p−1
k≥m
Z
{x:kxkp ≥pm }
j(kxkp ) dµp (x).
(8.13)
By (8.13), the equation (8.12) will get the following form p a(m) − a(m + 1) + a(m + 1) p−1 = a(m) − (p − 1)−1 a(m + 1) − a(m) }.
ψ(p−m+1 ) =
(8.14)
Let λ(m) be the eigenvalue of the Laplacian (ψ(D), D) corresponding to the ball B of radius pm . Then λ(m) = ψ(p−m+1 ) and the identity (8.14) gives the desired result, namely, the equation (8.11). Conversely, given a sequence {a(m)} as in (8.10), we define the sequence {λ(m)} by 8.11 and set Ψ(ξ) = ψ(kξkp ), J(x) = j(kxkp ),
where ψ(pm ) = λ(−m + 1),
and where j(p ) = a(m) − a(m + 1) (pm − pm−1 ) . m
(8.15)
It is straightforward to show that
Ψ(ζ) =
Z
Qp \{0}
1 − Rehx, ζi J(x) dµp (x) ,
whence Ψ is a non-negative definite function. It follows that the function exp(−t Ψ) is positive definite, whence it is the Fourier transform of a probability measure pt . Clearly, {pt }t>0 is a weakly continuous convolution semigroup of probability measures. By construction, each measure pt is rotation invariant. Finally, we can define the translation invariant Markov semigroup Pt f = f ∗ pt , as desired. Corollary 8.10 In the above notation the following statements are equivalent. (1) The sequence λ(m) is non-increasing. (2) The sequence ψ(pm ) is non-decreasing. 47
(3) The sequence j(pm ) is non-increasing. In particular, if the sequence a(m) is convex, then each of the equivalent properties (1)−(3) holds. Proof. The equivalence (1) ⇐⇒ (2) follows from the relation λ(m) = ψ(p−m+1 ). To prove that (1) ⇐⇒ (3), we apply (8.15) and obtain λ(m) − λ(m + 1) = (pm − pm−1 ) j(pm ) − j(pm+1 ) .
The equivalence (1) ⇔ (2) ⇔ (3) follows. Finally, (8.11) and the convexity of a (m) yield (1). Next, we consider strict monotonicity. Corollary 8.11 The following statements are equivalent (i) The sequence λ(m) is strictly decreasing, and λ(−∞) = +∞ . (ii) The sequence ψ(pm ) is strictly increasing, and ψ(+∞) = +∞ . R (iii) The sequence j(pm ) is strictly decreasing, and j(kxkp ) dµp (x) = +∞ . (iv) The associated rotation invariant Markov semigroup {Pt } is isotropic.
In particular, if the sequence a(m) is strictly convex and a(−∞) = +∞ , then each of the equivalent properties (i)–(iv) holds. Proof. The equivalence (i)⇔(ii)⇔(iii) follows by the same arguments as in the proof of Corollary 8.10. The convexity of a (m) together with a (−∞) = +∞ imply (i) following the same argument. We are left to show that (iv)⇐⇒(ii). Assume that {Pt } is a rotation invariant Markov semigroup as constructed in (1.2) – (1.4). The semigroup admits a continuous transition density p(t, x, y) = pt (x − y) with respect to the Haar measure µp ; the function pt is given by pt (y) =
Z
∞
qs (y) dσ t (s),
where qs (y) =
0
1 1Bs (0) (y) . µp Bs (0)
(8.16)
To find the Fourier transform pbt (ξ), we argue as follows. The ball Bs (0), pk ≤ s < pk+1 , is the compact subgroup p−k Zp of Qp , whence the measure ω s = qs µp coincides with the normed Haar measure of that compact subgroup. Since for any locally compact Abelian group, the Fourier transform of the normed Haar measure of any compact subgroup is the indicator of its annihilator group and, in our particular case, the annihilator of the group p−k Zp is the group pk Zp , we obtain ω cs (ξ) = 1pk Zp (ξ) = 1[0,p−k ] (kξkp ),
where pk ≤ s < pk+1 .
It follows that when kξkp = p−l ,
whence
pbt (ξ) =
X σ t (pk+1 ) − σ t (pk )t = σ t (pl+1 ) = exp −t ψ(kξkp ) ,
k:k≤l
ψ(p−l ) = log
48
1 . σ(pl+1 )
According to (2.3), the sequence σ(pl ) is assumed to be strictly increasing and to tend to zero as l → −∞ . Thus, ψ(pm ) is as claimed in (ii). Conversely, if a strictly increasing sequence ψ(pm ) as in (ii) is given, we define the strictly increasing sequence m −m+1 σ(p ) = exp −ψ(p ) .
Let σ : [0 , ∞) → [0 , 1) be any increasing bijection which takes the values σ(pm ) at the points pm . We define the function pt (y) by the equation (8.16). As σ(+∞) = 1, this is a probability density with respect to µp . It is straightforward that {pt }t>0 gives rise to a weakly continuous convolution semigroup of probability measures on Qp . Moreover, each pt is rotation invariant by construction. Thus, the semigroup Pt : f 7→ f ∗ pt is as desired. Remark 8.12 In [2], Albeverio and Karwowski started with a sequence {a(m)}m∈Z as in (8.10) and used the classic approach of backward and forward Kolmogorov equations to construct a Markov semigroup {Pt } on the ultra-metric measure space (Qp , dp , µp ). In particular, they showed [2, Theorem 3.2 9] that the Laplacian L of that semigroup has a pure point spectrum {λ(m)} as in (8.11), and the λ(m)-eigenspace is spanned by the functions fB , where B runs over all balls of diameter pm−1 . Our Theorem 8.9 shows that in fact the class of Markov semigroups constructed in [2] coincides with the class of rotation invariant Markov semigroups of this section, see Definition 8.8.
9
Product spaces and the Vladimirov and Taibleson Laplacians
Let {(Xi , di )}ni=1 be a finite sequence of ultra-metric spaces; we assume that all (Xi , di ) are separable and proper. Let (X, d) be their Cartesian product: X = X1 × ... × Xn and, for x = (xi ) and y = (yi ), we set d(x, y) = max {di (xi , yi ) : i = 1, 2, ..., n} . Thus (X, d) is a separable and proper ultra-metric space where each d-ball Br (a) ⊂ X is a product of di -balls Bri (ai ) ⊂ Xi of the same radius. Given a Radon measure µi on each (Xi , di ) N we define µ = µi on (X, d) and the scale of Lp -spaces Lp (X, m), p ≥ 1. Let D be the set of all compactly supported locally constant functions on the ultra-metric space (X, d) – the set of test functions. By the ultra-metric property, each function f ∈ D can be represented as a finite linear combination of indicators of d-balls. The set D is dense in all Lp , 1 ≤ p < ∞ , as well as in C0 – the space of all continuous functions vanishing at infinity. Consider the ultra-metric measure space (X, d, µ). According to the previous sections, there exists a rich class of isotropic Markov semigroups and corresponding Laplacians on (X, d, µ) as constructed in (1.2) – (1.4). Thanks to the product structure of (X, d, µ) one can define in a natural way a non-trivial and interesting class of Markov semigroups and Laplacians which are not isotropic. Namely, choosing on each (Xi , di , µi ) an isotropic Markov semigroup {Pit }, we define a Markov semigroup {P t } on (X, d, µ) as the tensor product of the {Pit }, t
P =
n O
Pit ,
i=1
or equivalently, in terms of the “one-dimensional” Laplacians Li corresponding to {Pit }, we set Lf (x) =
n X
Lixi f (x),
where f ∈ D
i=1
49
and x = (x1 , x2 , ..., xn ).
(9.1)
(The index xi refers to the action of Li on the i-th variable.) The semigroup {P t } admits a continuous transition density (heat kernel) p(t, x, y) with respect to the measure µ, n Y pi (t, xi , yi ). p(t, x, y) = i=1
The Laplacian (L, D) and the Dirichlet form (Lf, f ) restricted to D admit the following representation Z f (x) − f (y) J(x, dy), and Lf (x) = XZ Z 1 (f (x) − f (y))2 J(x, dy) dµ(x), (Lf, f ) = 2 X×X
where J(x, dy) =
n X
Ji (xi , yi ) dµi (yi ) ,
i=1
and each Ji (xi , yi ) is the jump-kernel associated with the Laplacian Li . In particular, we see that for each x ∈ X the measures J(x, dy) and µ(dy) are not mutually absolutely continuous (in the case when at least one of Xl is perfect, J(x, dy) is singular with respect to µ), whence the semigroup P t = exp(−tL) is not (!) an isotropic Markov semigroup, as constructed in (1.2) – (1.4) and studied in the previous sections. In this paper we do not intend to develop the general construction of products. Our aim is to study in detail two specific examples related to p-adic analysis. Namely, in the first example we consider the Vladimirov Laplacian which fits well to the general construction from above and therefore is not an isotropic Laplacian. In the second example, for comparison, we consider the Taibleson Laplacian defined in terms of the multidimensional Riesz kernels, see Taibleson [52] and Rodriguez-Vega and Zuniga-Galindo [47]. We show that the Taibleson Laplacian is isotropic, whence the general theory developed in the previous sections applies. This will allow us to improve the heat kernel bounds from [47] and to obtain some new results (transience/recurrence, independence on 1 ≤ p < ∞ of the Lp -spectrum, precise bounds of the moments of the corresponding Markov process etc.) We consider the linear space Qnp = Qp × ... × Qp (n times) over the field of p-adic numbers. The most natural ultra-metric on Qnp is dp (x, y) = kx − ykp , where n o kzkp = max kzi kp : i = 1, 2, ..., n . (9.2)
This is a norm on Qnp , that is,
kazkp = kakp kzkp ,
for all
a ∈ Qp , z ∈ Qnp .
N Let µp = µp,i be the additive Haar measure on the Abelian group Qnp and L2 = L2 (Qnp , µp ). Let D be the set of all compactly supported locally constant functions on the ultra-metric space (Qnp , dp ). Recall that D is a dense subset in L2 . The Vladimirov Laplacian. define the ultra-metric
For any given n-tuple α = (α1 , . . . , αn ) with entries αi > 0 we
n o dp,α (x, y) = max kxi − yi kαp i : i = 1, 2, ..., n . 50
The identity map
Qnp , dp,α → Qnp , dp
is a homeomorphism, but not bi-Lipschitz, unless αi = 1 for all i. This fact plays an essential role in the study of the class of Laplacians introduced next as a special instance of (9.1). Definition 9.1 Let α = (α1 , . . . , αn ). On the set D of test functions on (Qnp , dp ), we define the operator n X Dαxii f (x) , where x = (x1 , x2 , ..., xn ). Vα f (x) = i=1
Here,
Dαxii
is the p-adic fractional derivative of order αi acting on the i-th variable.
An operator of this type, namely the operator Vα on Q3p with α = (2, 2, 2), was introduced by Vladimirov [54] as an analogue of −∆, where ∆ is the classical Laplace operator in R3 . This operator, which we denote more briefly V2 , is translation invariant and homogeneous, that is, V2 τ y (f ) = τ y (V2 f ), where τ y f (x) = f (x + y). and V2 θa (f ) = kak2p θa (V2 f ),
where θ a f (x) = f (ax1 , ax2 , ax3 ).
It follows that the Green function G(x, y) of the operator V2 on Q3p is also translation invariant and homogeneous: G(x, y) = G(x − z, y − z)
and G(ax, ay) = G(x, y)/ kakp , a ∈ Qp .
In particular, setting E(x) = G(x, 0) , we obtain the identity E(a, a, a) =
E(1, 1, 1) . kakp
This identity was observed in [54]. It gives an idea of how the Green function of the operator V2 (in Vladimirov’s terminology, the fundamental solution of the equation V2 E = δ) behaves at infinity/at zero. Below, in Proposition 9.4, we will prove that E(a1 , a2 , a3 ) ≃
1 , kakp
(9.3)
where a = (a1 , a2 , a3 ) and kakp = max{kai kp }. We shall extend the asymptotic property (9.3) to the more general operators Vα . In general the homogeneity property is lost, whence we will develop some tools as a compensation. We list some properties of the operator (Vα , D) of Definition 9.1 which follow directly from the corresponding properties of the “one-dimensional Laplacians” Dαi . 1. (Vα , D) is a non-negative definite symmetric operator. 2. (Vα , D) admits a complete system of compactly supported eigenfunctions. In particular, the operator (Vα , D) is essentially self-adjoint.
51
3. The semigroup Pαt = exp(−t Vα ) is symmetric and Markovian. It admits a transition density (heat kernel) pα (t, x, y) which has the following form pα (t, x, y) =
n Y
pαi (t, xi , yi ).
i=1
In particular, for all x ∈ Qnp and all t > 0,
c1 t−A ≤ pα (t, x, x) ≤ c2 t−A , where A=
n X 1 αi i=1
and c1 , c2 > 0 are some constants.
4. The semigroup {Pαt }t>0 is transient if and only if A > 1. 5. For all u ∈ D the following identities hold Z α u(x) − u(y) Jα (x, dy) V u(x) =
and
Qn p
α
(V u, u) =
1 2
ZZ
n Qn p ×Qp
where Jα (x, dy) =
n X
2 u(x) − u(y) Jα (x, dy) dµp (x) ,
Jαi (xi − yi ) dµp,i (yi )
with
i1
Jαi (xi − yi ) =
1 pαi − 1 . i 1 − p−αi −1 kxi − yi k1+α p
In particular, the semigroup Pαt = exp(−t Vα ) is not an isotropic Markov semigroup, as constructed in (1.2) – (1.4). Our next goal is to find off-diagonal estimates of the heat kernel pα (t, x, y) and, in the transient case, the Green function Gα (x, y) in metric terms. Observe that thanks to the group structure of Qnp , the functions (x, y) 7→ pα (t, x, y) and (x, y) 7→ Gα (x, y) are translation invariant. Hence, setting pα (t, z) = pα (t, z, 0) and Gα (z) = Gα (z, 0) , we obtain pα (t, x, y) = pα (t, x − y) and Gα (x, y) = Gα (x − y). Proposition 9.2 The heat kernel satisfies n Y
(
t1+1/αi min 1, pα (t, z) ≃ t−A i kzi k1+α p i=1 uniformly for all t > 0 and z ∈ Qnp .
)
In particular, there are c1 , c2 > 0 such that for all t > kzkp,α , c1 t−A ≤ pα (t, z) ≤ c2 t−A 52
(9.4)
Proof. We recall the definition of kzkp,α and apply Theorem 8.3: the “one-dimensional” heat kernels can be estimated as follows. ) ( 1 t1+1/αi t . pαi (t, zi ) ≃ 1+αi ≃ 1/αi min 1, i t kzi k1+α t1/αi + kzi kp p
This implies the proposed result.
Proposition 9.3 Assume that the semigroup {Pαt }t>0 is transient. Then, for all z ∈ Qnp and some C1 > 0, A−1 1 α G (z) ≥ C1 . kzkp,α For any κ > 0, we define the set o n o n αi αi n . Ω(κ) = x ∈ Qp : max kxi kp ≤ κ min kxi kp i
i
Then, for all z ∈ Ω(κ) and some constant C2 > 0 which depends on κ, A−1 1 α G (z) ≤ C2 . kzkp,α Proof. Recall that transience holds precisely when A > 1. To prove the lower bound, we use (9.4) and write !A−1 Z ∞ Z ∞ Z ∞ 1 −A α pα (t, z) dt ≥ C1 t dt = c1 . pα (t, z) dt ≥ G (z) = kzkp,α kzkp,α kzkp,α 0 On the other hand we have Gα (z) =
Z
kzkp,α 0
+
Z
∞
kzkp,α
!
pα (t, z) dt = I + II .
To estimate the second term II , we use again the inequality (9.4), A−1 Z ∞ 1 −A t dt ≃ II ≃ . kzkp,α kzkp,α To estimate the first term we use Proposition 9.2, Z kzk n Y p,α t1+1/αi t−A I ≤ c 1+αi dt kz k 0 i p i=1 Z kzk Y n n Y p,α 1 1 n+1 n ′ = c t dt = c 1+αi 1+αi kzkp,α . kzi kp kzi kp 0 i=1 i=1 When z ∈ Ω(κ), we obtain I
≤ c′′
n n+1 Y 1 1 αi αi ′′ min kz k ≤ c min kz k i p i p 1+αi kz kzi kp i kp i=1 i=1
n Y
n Y αi = c min kzi kp ′′
1
. (kzi kαp i )1/αi i=1 53
Next,
n Y i=1
1 (kzi kαp i )1/αi
≤
n Y i=1
whence
1
1/αi = min kzj kαp j ′′
I ≤c
1 min kzj kαp j
Again using the fact that z ∈ Ω(κ), we write 1 min kzj kαp j
!A−1
≤
κ max kzj kαp j
1 min kzj kαp j
A
,
A−1 .
!A−1
1 kzkp,α
= c(κ)
!A−1
.
The obtained upper bounds on the integrals I and II imply the desired upper bound for Gα (z). Proposition 9.4 Let α = (α1 , . . . , αn ) = (β, . . . , β) be an n-tuple having all entries equal to β. Assume that (n − 1)/2 < β < n. Then the semigroup {Pαt } is transient and the Green function Gα (z) satisfies A−1 A−1 1 1 α c1 ≤ G (z) ≤ c2 , (9.5) kzkp,α kzkp,α equivalently, c1
1 kzkp
n−β
α
≤ G (z) ≤ c2
1 kzkp
n−β
,
(9.6)
for all z ∈ Qnp and some c1 , c2 > 0. Proof. Transience follows from (9.4) because A = n/β > 1. Then Proposition 9.3 implies the desired lower bound of the Green function A−1 1 α . G (z) ≥ c1 kzkp,α To prove the upper bound, we observe that like V2 , the Laplacian Vα is homogeneous, that is Vα ◦ θa = kakβp · θa ◦ Vα , for all a ∈ Qp . This implies that also the Green function Gα (z) is homogeneous, that is Gα (az) = kakpn−β Gα (z) for all a ∈ Qp and z ∈ Qnp . Assume now that kzkp,α = kz1 kβp > 0. Then Gα (z) = Gα z1 (1, z2 /z1 , ..., zn /z1 ) = kz1 kpn−β Gα (1, z2 /z1 , ..., zn /z1 ) A−1 1 = Gα (1, z2 /z1 , ..., zn /z1 ) kzkp,α A−1 1 sup Gα (1, x2 , ..., xn ) : xi ∈ Zp . ≤ kzkp,α 54
Next we apply our assumption β > (n − 1)/2 and the heat kernel upper bound resulting from Proposition 9.2 Z ∞ α G (1, x2 , ..., xn ) = pα (t, (1, x2 , ..., xn ) dt 0 Z 1 Z ∞ + pα (t, (1, x2 , ..., xn ) dt = 1 0 Z 1 Z ∞ − n 1+ 1 −n t β t β dt + c′ ≤ c t β dt = c2 < ∞. 0
1
The desired upper bound follows. The Taibleson Laplacian. The Fourier transform F : f 7→ fb on the locally compact Abelian group Qnp is a linear isomorphism from D onto itself. This basic fact justifies the following Definition (cf. Definition 8.1). Definition 9.5 The Taibleson operator Tα for α > 0 is defined on D as α f (ζ) = kζkα fb(ζ), Td p
ζ ∈ Qnp .
It follows that (Tα , D) is an essentially self-adjoint and non-negative definite operator in L2 . This operator and the associated semigroup Pαt = exp(−t Tα ) were introduced and studied by Rodriguez-Vega and Zuniga-Galindo [47].In particular, it was shown that Z f (x) − f (y) pα − 1 α dµp (y). (9.7) T f (x) = −α−n 1−p kx − ykα+n Qn p p The equation (9.7) shows that the operator (−Tα , D) satisfies the max-principle, whence its semigroup {Pαt } is Markovian. Our aim is to show that (Pαt ) is an isotropic Markov semigroup on the ultra-metric measure space (Qnp , dp , µp ). Our first observation is that the spectrum of the symmetric operator (Tα , D) coincides with the range of the function ζ 7→ kζkαp , Spec(Tα ) = {pkα : k ∈ Z} ∪ {0}. The eigenspace H(λ) of the operator (Tα , D) corresponding to the eigenvalue λ = pkα, is spanned by the functions 1 1 1k n− 1 k−1 n fk = µp (pk Znp ) p Zp µp (pk−1 Znp ) p Zp and all its shifts fk (· + a) with a ∈ Qnp /pk Znp . Indeed, computing the Fourier transform of the function fk ,
we obtain
fbk (ζ) = 1{kζkp ≤pk } − 1{kζkp ≤pk−1 } = 1{kζkp =pk } , α f (ζ) = kζkα fb (ζ) = pkα fb (ζ). [ T k k p k
All the above shows that the semigroup {Pαt } is an isotropic Markov semigroup defined on the ultra-metric measure space (Qnp , dp , µp ) as constructed in (1.2) – (1.4) and studied in the previous sections. Namely, we have Z Pαt f (x) =
∞
0
Qr f (x) dσ tα (r) ,
55
where, as in §12, σ α (r) =
exp − 0
p α r
if r > 0 , . if r = 0 .
Definition 8.8 of a rotation invariant Laplacian on Qp can be carried over to Qnp . The Taibleson operator Tα is an example of a rotation invariant Laplacian. Theorem 8.9, Corollary 8.10 and Corollary 8.11 and their proofs remain valid also for Qnp . Here we provide a short proof of a slightly weaker result that is of significance for us. Theorem 9.6 The equation (L, D) = (ψ(T), D), where ψ is an arbitrary increasing bijection [0 , ∞) → [0 , ∞), gives a complete description of the class of isotropic Laplacians on the ultrametric measure space (Qnp , dp , µp ). Proof. Let ψ : [0, ∞) 7→ [0, ∞) be an increasing bijection. Let T : = T1 be the Taibleson Laplacian. By Theorem 7.5, the operator (ψ(T), D) defined on the ultra-metric measure space (Qnp , dp , µp ) is an isotropic Laplacian. Conversely, let (L, D) be an isotropic Laplacian on (Qnp , dp , µp ). Let dp∗ be the intrinsic distance associated with L. By construction, dp∗ is an increasing function of dp , see (1.6). Since the range of dp is the set {pk : k ∈ Z} ∪ {0}, one can choose an increasing bijection ϕ : [0 , ∞) → [0 , ∞) such that dp∗ = φ(dp ). Let λ(B) and τ (B) be the eigenvalues of (L, D) and (T, D), respectively, corresponding to the ball B ⊂ Qnp . Since the intrinsic distance associated with T is p−1 dp , we get λ(B) = =
1 1 = diamp∗ (B) ϕ diamp (B) 1 =: ψ τ (B) , ϕ p/τ (B)
where ψ(s) = 1/φ(p/s), an increasing bijection of [0 , ∞) onto itself. Since both (L, D) and (ψ(T), D) are isotropic Laplacians defined on the ultra-metric measure space (Qnp , dp , µp ) whose sets of eigenvalues coincide, we get (L, D) = (ψ(T), D), or equivalently, in terms of the Fourier transform, c (ζ) = ψ(kζk ) fb(ζ), Lf p
for all f ∈ D and ζ ∈ Qnp , which finishes the proof. Returning to the Tailbleson operator, observe that the intrinsic ultra-metric of Tα is kx − ykp α dp∗ (x, y) = . p The spectral distribution function Nα (x, τ ) = Nα (τ ) is the non-decreasing, left-continuous staircase function which has jumps at the points τ k = pkα , k ∈ Z, and takes values Nα (τ k ) = p(k−1)n at these points. It follows that p−n τ n/α ≤ Nα (τ ) < τ n/α . In particular, τ 7→ Nα (τ ) is a doubling function, and Corollary 4.5 implies the following result. 56
Theorem 9.7 The semigroup {Pαt }t>0 on Qnp admits a continuous transition density pα (t, x, y) with respect to the Haar measure µp which satisfies c1 t t1/α + kx − ykp
n+α ≤ pα (t, x, y) ≤
c2 t t1/α + kx − ykp
n+α ,
(9.8)
for some constants c1 , c2 > 0. In particular, the semigroup {Pαt }t>0 is transient if and only if α < n. In the transient case, the Green function – Taibleson’s Riesz kernel – is Gα (x, y) =
1 1 − p−α . 1 − pα−n kx − ykpn−α
The upper bound in (9.8) was proved in [47].
10
Random walks on a tree and jump processes on its boundary
Rooted trees and their boundaries. A tree is a connected graph T without cycles (closed paths of length ≥ 3). We tacitly identify T with its vertex set, which is assumed to be infinite. We write u ∼ v if u, v ∈ T are neighbors. For any pair of vertices u, v ∈ T , there is a unique shortest path, called geodesic segment π(u, v) = [u = v0 , v1 , . . . , vk = v] such that vi−1 ∼ vi and all vi are disctinct. If u = v then this is the empty or trivial path. The number k is the length of the path (the graph distance between u and v). In T we choose and fix a root vertex o. We write |v| for the length of π(o, v). The choice of the root induces a partial order on T , where u ≤ v when u ∈ π(o, v). Every v ∈ T \ {o} has a unique predecessor v − = vo− with respect to o, which is the unique neighbour of v on π(o, v). Thus, the set of all (unoriented) edges of T is E(T ) = {[v − , v] : v ∈ T , v 6= o} . For u ∈ T , the elements of the set {v ∈ T : v − = u} are the successors of u, and its cardinality deg+ (u) is the forward degree of u. In this and the next section, we assume that 2 ≤ deg+ (u) < ∞
for every u ∈ T . o
• .... ... ... ... ..... ... ... . . ... . ... ... ... .. ... ... . . ... .. . ... . .. ... . . ... .. . . ... .. . ... . .. ... . . ... .. . . ... .. . ... . ..... . ... • . .. .... ... ... . . ... ..... ... . .. . . . . ... ... . . .. . . . ... . . ... . ... ... ... ... ... ... ... ... ... ... ... . ... . . .. . . ..... ...... ..... • . ..... . . . . . .. .... ... ... .. .... .. .... . . . . . . ... ... ... ..... . ... . .. . . . . ... . . . . . . .. .. ... ... .... .... .. .. ... ....... ...... ... ... ...... ...... ..• ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
u∧v
v
u ∂T
.. .
Figure 4 57
(10.1)
A (geodesic) ray in T is a one-sided infinite path π = [v0 , v1 , v2 , . . . ] such that vn−1 ∼ vn and all vn are disctinct. Two rays are equivalent if their symmetric difference (as sets of vertices) is finite. An end of T is an equivalence class of rays. We shall typically use letters x, y, z to denote ends (and letters u, v, w for vertices). The set of all ends of T is denoted ∂T . This is the boundary at infinity of the tree. For any u ∈ T and x ∈ ∂T , there is a unique ray π(u, x) which is a representative of the end (equivalence class) x and starts at u. We write Tb = T ∪ ∂T.
For u ∈ T , the branch of T rooted at u is the subtree Tu that we identify with its set of vertices Tu = {v ∈ T : u ≤ v} ,
(10.2)
so that To = T . We write ∂Tu for the set of all ends of T which have a representative path contained in Tu , and Tbu = Tu ∪ ∂Tu . For w, z ∈ Tb, we define their confluent w ∧ z = w ∧o z with respect to the root o by the relation π(o, w ∧ z) = π(o, w) ∩ π(o, z) . It is the last common element on the geodesics π(o, w) and π(o, z), a vertex of T unless w = z ∈ ∂T . See Figure 4. One of the most common ways to define an ultra-metric on Tb is ( 0, if z = w , (10.3) de (z, w) = −|z∧w| e , if z 6= w . Then Tb is compact, and T is open and dense. We are mostly interested in the compact ultrametric space ∂T . In the metric de of (10.3), each de -ball with centre x ∈ ∂T is of the form ∂Tu for some x ∈ π(o, x). Indeed ∂Tu = Be−|u| (x)
for every o ∈ π(o, x) ,
and
Λde (x) = {e−|u| : u ∈ π(o, u)} .
Conversely, we now start with a compact ultra-metric space (X, d) that does not possess isolated points, and construct a tree T as follows: The vertex set of T is the collection B = {Br (x) : x ∈ X , r > 0} of all closed balls in (X, d), already encountered in §7. Here, we may assume (if we wish) that r ∈ Λd (x). We now consider any ball v = B ∈ B as a vertex of a tree T . We choose our root vertex as o = X, which belongs to B by compactness. Neighborhood is given by the predecessor relation of balls, as given by Definition 2.1. That is, if v = B then u = B ′ is the predecessor vertex v − of v in the tree T . By compactness, each x has only finitely many successors, and since there are no isolated points in X, every vertex has at least 2 successors, so that (10.1) holds. This defines the tree structure. For any x ∈ X, the collection of all balls Br (x), r ∈ Λd (x), ordered decreasingly, forms the set of vertices of a ray in T that starts at o. Via a straightforward exercise, the mapping that associates to x the end of T represented by that ray is a homeomorphism from X onto ∂T . Thus, we can identify X and ∂T as ultra-metric spaces. 58
In this identification, if originally a vertex u was interpreted as a ball Br (x), r ∈ Λd (x), then the set ∂Tu of ends of the branch Tu just coincides with the ball Br (x). That is, we are identifying each vertex u of T with the set ∂Tu . If we start with an arbitrary locally finite tree and take its space of ends as the ultra-metric space X, then the above construction does not recover vertices with forward degree 1, so that in general we do not get back the tree we started with. However, via the above construction, the correspondence between compact ultra-metric spaces without isolated points (perfect ultrametric spaces) and locally finite rooted trees with forward degrees ≥ 2 is bijective. It is well known that any ultra-metric space X which is both compact and perfect is homeomorphic to the ternary Cantor set C ⊂ [0, 1]. When X is not compact but still perfect we have a homeomorphism X ≃ C \ {p}, where p ∈ C is any fixed point. For the rest of this and the next section, we shall abandon the notation X for compact and perfect ultra-metric space. We consider X as the boundary ∂T of a locally finite, rooted tree with forward degrees ≥ 2. At the end, we shall comment on how one can handle the presence of vertices with forward degree 1, as well as the non-compact case. There are many ways to equip ∂T with an ultra-metric that has the same topology and the same compact-open balls ∂Tx , x ∈ T , possibly with different radii than in the standard metric (10.3). The following is a kind of ultra-metric analogue of a length element. Definition 10.1 Let T be a locally finite, rooted tree T with deg+ (x) ≥ 2 for all x. An ultra-metric element is a function φ : T → (0 , ∞) with (i) φ(v − ) > φ(v) (ii)
lim φ(vn ) = 0
for every v ∈ T \ {o} , along every geodesic ray π = [v0 , v1 , v2 , . . . ] .
It induces the ultra-metric dφ on ∂T given by ( 0, dφ (x, y) = φ(x ∧ y) ,
if x = y , if x 6= y .
The balls in this ultra-metric are again the sets ∂Tu = Bφ(u) (x) , x ∈ ∂Tu . Note that condition (ii) in the definition is needed for having that each end of T is non-isolated in the metric dφ . The metric de of (10.3) is of course induced by φ(x) = e−|x| . Lemma 10.2 For a tree as in Definition 10.1, every ultra-metric on ∂T whose closed balls are the sets ∂Tu , u ∈ T , is induced by an ultra-metric element on T . Proof. Given an ultra-metric d as stated, we set φ(v) = diam(∂Tv ), the diameter with respect to the metric d. Since deg+ (v − ) ≥ 2 for any v ∈ T \ {o}, the ball ∂Tv− is the disjoint union of at least two balls ∂Tu with u− = v − . Therefore we must have diam(∂Tv ) < diam(∂Tv− ), and property (i) holds. Since no end is isolated, φ satisfies (ii). It is now straightforward that dφ = d.
59
In view of this correspondence, in the sequel we shall replace the subscript d referring to the metric d = dφ by the subscript φ referring to the ultra-metric element. We note that Λφ (x) = {φ(u) : u ∈ π(o, x)}
diamφ (∂T ) = φ(o) ,
and Λφ = {φ(v) : v ∈ T }.
We also note here that for any x ∈ ∂T and v ∈ π(o, x), the balls with respect to dφ are ( ∂Tv for φ(v) ≤ r < φ(v − ) , if v 6= o Br (x) = Brφ (x) = ∂T for r ≥ φ(o) , if v = o .
(10.4)
(10.5)
Isotropic jump processes on the boundary of a tree. In view of the explanations given above, we can consider the isotropic jump processes of (1.2)–(1.4) on X = ∂T . Since this space is compact, we may assume that the reference measure µ is a probability measure on ∂T . Given µ, a distance distribution σ with properties (2.3), and an ultra-metric element φ on T , we can now refer to the (dφ , µ, σ)-process simply as the (φ, µ, σ)-process on ∂T . We can write the semigroup and its transition probabilities in detail as follows. For x ∈ ∂T and π(0, x) = [o = v0 , v1 , v2 , . . . ], using (10.5), P t f (x) =
∞ X
n=0
ctn Pφ(vn ) f (x) ,
where ct0 = 1 − σ t φ(v0 )
and ctn = σ t φ(vn−1 ) − σ t φ(vn ) for n ≥ 1 .
Thus, for arbitrary u ∈ T and x ∈ ∂T as above
P[Xt ∈ ∂Tu | X0 = x] =
∞ X
n=0
ctn
µ(∂Tvn ∩ ∂Tu ) . µ(∂Tvn )
(10.6)
We know that we have some freedom in the choice of the measure σ: any two measures whose distribution functions coincide on the value set Λφ of φ give rise to the same process. Recall the Definition 1.4 of the standard (d, µ)-process, now to be re-named the standard (φ, µ)-process. Nearest neighbour random walks on a tree. On a tree as a discrete structure, there are other, very well studied stochastic processes, namely random walks. Our aim is to analyze how they are related with isotropic jump processes on the boundary of the tree. A good part of the material outlined next is taken from the book of Woess [60]. An older, recommended reference is the seminal paper of Cartier [15]. A nearest neighbour random walk on the locally finite, infinite tree T is induced by its stochastic transition matrix P = p(u, v) u,v∈T with the property that p(u, v) > 0 if and only if u ∼ v. The resulting discrete-time Markov chain (random walk) is written (Zn )n≥0 . Its n-step transition probabilities p(n) (u, v) = Pu [Zn = v], u, v ∈ T, are the elements of the nth power of the matrix P. The notation Pu refers to the probability measure on the trajectory space that governs the random walk starting at u. We assume that the random walk is transient, i.e., with probability 1 it visits any finite set only finitely often. Thus, 0 < G(u, v) < ∞ for all u, v ∈ T , where G(u, v) =
∞ X
n=0
60
p(n) (u, v)
is the Green kernel of the random walk. In addition, we shall also make crucial use of the quantities F (u, v) = Pu [Zn = v for some n ≥ 0]
and U (v, v) = Pv [Zn = v for some n ≥ 1] .
We shall need several identities relating them and start with a few of them, valid for all u, v ∈ T . G(u, v) = F (u, v)G(v, v) 1 G(v, v) = 1 − U (v, v) X U (v, v) = p(v, u)F (u, v)
(10.7) (10.8) (10.9)
u
F (u, v) = F (u, w)F (w, v)
whenever w ∈ π(u, v)
(10.10)
The first three hold for arbitrary denumerable Markov chains, while (10.10) is specific for trees (resp., a bit more generally, when w is a so-called cut point between u and v). The identities show that those quantities are completely determined just by all the F (u, v), where u ∼ v. More identities, as to be found in [60, Chapter 9], will be displayed and used later on. By transience, the random walk Zn must converge to a random end, a simple and well-known fact. See e.g. [15] or [60, Theorem 9.18]. Lemma 10.3 There is a ∂T -valued random variable Z∞ such that for every starting point u ∈ T, Pu [Zn → Z∞ in the topology of Tb] = 1.
In brief, the argument is as follows: by transience, random walk trajectories must accumulate at ∂T almost surely. If such a trajectory had two distinct accumulation points, say x and y, then by the nearest neighbour property, the trajectory would visit the vertex x ∧u y infinitely often, which can occur only with probability 0. We can consider the family of limit distributions ν u , u ∈ T , where for any Borel set B ⊂ ∂T , ν u (B) = Pu [Z∞ ∈ B] . The sets ∂Tu , u ∈ T (plus the empty set), form a semi-algebra that generates the Borel σ-algebra of ∂T . Thus, each ν u is determined by the values of those sets. There is an explicit formula, compare with [15] or [60, Proposition 9.23]. For v 6= o, 1 − F (v, v − ) F (u, v) , if u ∈ {v} ∪ (T \ Tv ) , 1 − F (v − , v)F (v, v − ) (10.11) ν u (∂Tv ) = − ) − F (v − , v)F (v, v − ) F (v, v , if u ∈ Tv . 1 − F (u, v) 1 − F (v − , ν)F (v, v − ) A harmonic function is a function h : T → R with Ph = h, where X P h(u) = p(u, v)h(v) . v
For any Borel set B ⊂ ∂T , the function u 7→ ν u (B) is a bounded harmonic function. One deduces that all ν u are comparable: p(k) (u, v) ν u ≤ ν v , where k is the length of π(u, v). Thus, for any function ϕ ∈ L1 (∂T, ν o ), the function hϕ defined by Z ϕ dν u hϕ (u) = ∂T
61
is finite and harmonic on T . It is often called the Poisson transform of ϕ. We next define a measure m on T via its atoms: m(o) = 1, and for v ∈ T \ {o} with π(o, v) = [o = v0 , v1 , . . . , vk = v], m(v) =
p(v0 , v1 )p(v1 , v2 ) · · · p(vk−1 , vk ) . p(v1 , v0 )p(v2 , v1 ) · · · p(vk , vk−1 )
(10.12)
Then for all u, v ∈ T , m(u)p(u, v) = m(v)p(v, u) ,
and consequently
m(u)G(u, v) = m(v)G(v, u) ;
(10.13)
the random walk is reversible. This would allow us to use the electrical network interpretation of (T, P, m), for which there are various references: see e.g. Yamasaki [62], Soardi [51], or – with notation as used here – [60, Chapter 4]. We do not go into its details here; each edge e = [v − , v] ∈ E(T ) is thought of as an electric conductor with conductance a(v − , v) = m(v)p(v, v − ). We get the Dirichlet form ET = ET,P for functions f, g : T → R, defined by X f (v) − f (v − ) g(v) − g(v − ) a(v − , v) . ET (f, g) =
(10.14)
[v− ,v]∈E(T )
It is well defined for f, g in the space D(T ) = D(T, P) = {f : T → R | ET (f, f ) < ∞}. Harmonic functions of finite energy and their boundary values. the subspace HD(T ) = HD(T, P) = {h ∈ D(T, P) : Ph = h}
(10.15) We are interested in
of harmonic functions with finite power. The terminology comes from the interpretation of such a function as the potential of an electric flow (or current), and then ET (h, h) is the power of that flow.1 Every function in HD(T, P) is the Poisson transform of some function ϕ ∈ L2 (∂T, ν o ). This is valid not only for trees, but for general finite range reversible Markov chains, and follows from the following facts. 1. Every function in HD is the difference of two non-negative functions in HD. 2. Every non-negative function in HD can be approximated, monotonically from below, by a sequence of non-negative bounded functions in HD. 3. Every bounded harmonic function (not necessarily with finite power) is the Poisson transform of a bounded function on the boundary. In the general setting, the latter is the (active part of) the Martin boundary, with ν u being the limit distribution of the Markov chain, starting from u, on that boundary. (1) and (2) are 1
In the mathematical literature, mostly the expression “energy” is used for ET (h, h), but it seems that “power” is the more appropriate terminology from Physics.
62
contained in [62] and [51], while (3) is part of general Martin boundary theory, see e.g. [60, Theorem 7.61]. Thus, we can introduce a form EHD on ∂T by setting D(∂T, P) = {ϕ ∈ L1 (∂T, ν o ) : ET (hϕ , hϕ ) < ∞} , EHD (ϕ, ψ) = ET (hϕ , hψ ) for ϕ, ψ ∈ D(∂T, P).
(10.16)
Jump processes on the boundary of a tree. Kigami [34] elaborates an expression for the form EHD (ϕ, ψ) of (10.16) by considerable effort, shows its regularity properties and then studies the jump process on ∂T induced by this Dirichlet form. We call this the boundary process associated with the random walk. Now, there is a rather simple expression for EHD . We define the Na¨ım kernel on ∂T × ∂T by m(o) , if x 6= y , (10.17) Θo (x, y) = G(o, o)F (o, x ∧ y)F (x ∧ y, o) +∞ , if x = y .
In our case, m(o) = 1, but we might want to change the base point, or normalize the measure m in a different way.
Theorem 10.4 For any transient nearest neighbour random walk on the tree T with root o, and all functions ϕ, ψ in D(∂T, P), Z Z 1 ϕ(x) − ϕ(y) ψ(x) − ψ(y) Θo (x, y) dν o (x) dν o (y) . EHD (ϕ, ψ) = 2 ∂T ∂T
There is a general definition of the Na¨ım kernel [41] that involves the Martin boundary, which in the present case is ∂T . A proof of Theorem 10.4 is given in [21] in a setting of abstract potential theory on Green spaces, which are locally Euclidean. The definition of [41] refers to the same type of setting. Now, infinite networks, even when seen as metric graphs, are not locally Euclidean. In this sense, so far the definition of the kernel and a proof of Theorem (10.4) for transient, reversible random walks have not been well accessible in the literature. In a forthcoming paper, Georgakopoulos and Kaimanovich [25] will provide those “missing links”. We shall give a direct and simple proof of Theorem 10.4 for the specific case of trees. We start with the following observation. Lemma 10.5 The measure Θo (x, y) dν o (x) dν o (y) on ∂T ×∂T is invariant with respect to changing the base point (root) o.
Proof. We want to replace the base point o with some other u ∈ T . We may assume that u ∼ o. Indeed, then we may step by step replace the current base point by one of its neighbors to obtain the result for arbitrary u. Recall that the confluent that appears in the definition (10.17)) of Θo depends on the root o, while for Θx it becomes the one with respect to x as the new root. It is a well-known fact that G(u, u ∧o x) dν u (x) = K(u, x) = , dν o G(o, u ∧o x) the Martin kernel. Thus, we have to show that for all x, y ∈ ∂T (x 6= y) m(u)K(u, x)K(u, y) m(o) = . G(o, o)F (o, x ∧o y)F (x ∧o y, o) G(u, u)F (u, x ∧u y)F (x ∧u y, u) 63
Case 1. x, y ∈ ∂Tu . Then x ∧o y = x ∧u y =: v ∈ Tu , and u ∧o x = u ∧o y = u. Thus, using (10.7), (10.10) and the fact that by (10.13) m(u)/G(o, u) = m(o)/G(u, o), we obtain m(u)K(u, x)K(u, y) G(u, u)F (u, x ∧u y)F (x ∧u y, u)
= = = =
m(u) G(u, u) 2 G(u, u)F (u, v)F (v, u) G(o, u) m(o)G(u, u) F (u, v)F (v, u)G(o, u)G(u, o) m(o) F (u, v)F (v, u)F (o, u)F (u, o)G(o, o) m(o) , F (o, v)F (v, o)G(o, o)
as required. There are 3 more cases. Case 2. x, y ∈ ∂T \ ∂Tu . Then x ∧o y = x ∧u y =: w ∈ T \ Tu ,
and u ∧o x = u ∧o y = o.
Case 3. x ∈ ∂Tu , y ∈ ∂T \ ∂Tu . Then x ∧o y = o, x ∧u y = u, u ∧o x = u
and u ∧o y = o.
Case 4. x ∈ ∂T \ ∂Tu , y ∈ ∂Tu . This is like Case 3, exchanging the roles of x and y. In all cases 2–4, the computation is done very similarly to Case 1, a straightforward exercise. For proving Theorem 10.4, we need a few more facts related with the network setting; compare e.g. with [60, §4.D]. The space D(T ) of (10.15) is a Hilbert space when equipped with the inner product (f, g) = ET (f, g) + f (o)g(o) . The subspace D0 (T ) is defined as the closure of the space of finitely supported functions in D(T ). It is a proper subspace if and only if the random walk is transient, and then the function Gv (u) = G(u, v) is in D0 (T ) for any v ∈ T [62], [51]. We need the formula ET (f, Gv ) = m(v)f (v)
for every f ∈ D0 (T ) .
(10.18)
Given a branch Tw of T (w ∈ T \ {o}), we can consider it as a subnetwork equipped with the same conductances a(u, v) for [u, v] ∈ E(Tw ). The associated measure on Tw is ( X m(u) if u ∈ Tw \ {w} , mTw (u) = a(u, v) = − m(w) − a(w, w ) if u = w . v∈T :v∼u z
The resulting random walk on Tw has transition probabilities if u ∈ Tw \ {w} , v ∼ u , p(u, v) a(v, w) pTw (u, v) = = p(w, v) mTw (u) if u = w , v ∼ u . 1 − p(w, w− ) 64
We have FTw (u, u− ) = F (u, u− ) and thus also FTw (u, w) = F (u, w) for every u ∈ Tw \ {w}, because before its first visit to w, the random walk on Tw obeys the same transition probabilities as the original random walk on T . It is then easy to see [60, p. 241] that the random walk on Tw is transient if and only if for the original random walk, F (w, w− ) < 1, which in turn holds if and only if ν o (∂Tw ) > 0. (In other parts of this and the preceding two sections, this is always assumed, but for the proof of Theorem 10.4, we just assume the random walk on the whole of T to be transient.) Conversely, if F (w, w− ) = 1 then F (u, w) = 1 for all u ∈ Tw . Below, we shall need the following formula for the limit distributions. Lemma 10.6 For u ∈ T \ {o},
Proof. By (11.7),
ν u (∂Tu ) = 1 − p(u, u− ) G(u, u) − G(u− , u) . G(u, u)p(u, u− ) =
F (u, u− ) 1 − F (u, u− )F (u− , u)
Thus, p(u, u− ) G(u, u) − G(u− , u) = 1 − F (u− , u) G(u, u)p(u, u− ) = 1 − ν u (∂Tu )
after a short computation using (10.11)
Proof of Theorem 10.4. We first prove the Doob-Na¨ım formula (shortly, D-N-formula) for the case when ϕ = 1∂Tv and ψ = 1∂Tw for two proper branches Tv and Tw of T . They are either disjoint, or one of them contains the other. Case 1. Tw ⊂ Tv . (The case Tv ⊂ Tw is analogous by symmetry.) This means that w ∈ Tv . For x, y ∈ ∂T we have ϕ(x) − ϕ(y) ψ(x) − ψ(y) = 1
if x ∈ ∂Tw and y ∈ ∂T \ ∂Tv or conversely, and = 0 otherwise. By Lemma 10.5, we may choose v as the base point. Thus, the right hand side of the identity is Z Z m(v) ν v (∂T \ ∂Tv )ν v (∂Tw ) , Θv (x, y) dν v (x) dν v (y) = G(v, v) ∂T \∂Tv ∂Tw since x ∧v y = v and F (v, v) = 1. Let us now turn to the left hand side of the D-N-formula. The Poisson transforms of ϕ and ψ are hϕ (u) = ν u (∂Tv ) and hψ (u) = ν u (∂Tw ). By (10.11), hϕ (u) = F (u, v)ν v (∂Tv ) , u ∈ {v} ∪ (T \ Tv ) 1 − hϕ (u) = F (u, v)(∂T \ ∂Tv ) , u ∈ Tv . We set Fv (u) = F (u, v) and write hϕ (u) − hϕ (u− ) = 1 − hϕ (u− ) − 1 − hϕ (u) 65
whenever this is convenient, and analogously for hψ . Then we get ET (hϕ , hψ ) X =
a(u, u− ) F (u, v) − F (u− , v) ν v (∂Tv ) F (u, v) − F (u− , v) ν v (∂Tw )
[u,u− ]∈E(T )\E(Tv )
−
X
a(u, u− ) F (u, v) − F (u− , v) ν v (∂T \ ∂Tv ) F (u, v) − F (u− , v) ν v (∂Tw )
[u,u− ]∈E(Tv )\E(Tw )
+
X
a(u, u− ) F (u, v) − F (u− , v) ν v (∂T \ ∂Tv ) F (u, v) − F (u− , v) ν v (∂T \ ∂Tw )
[u,u− ]∈E(Tw )
= ET (Fv , Fw )ν v (∂Tv )ν v (∂Tw ) − ETv (Fv , Fw )ν v (∂Tw ) + ETw (Fv , Fw )ν v (∂T \ ∂Tv ) ,
where of course ETv is the Dirichlet form of the random walk on the branch Tv , as discussed above, and analogously for ETw . Now Fv = Gv /G(v, v) by (10.7), whence (10.18) yields ET (Fv , Fw ) =
m(v)F (v, w) ET (Gv , Fw ) = . G(v, v) G(v, v)
(10.19)
Recall that for the random walk on Tv , we have FTv (u, v) = F (u, v) for every u ∈ Tv . Also, mTv (v) = m(v) − a(v, v − ) = m(v) 1 − p(v, v − ) .
We apply (10.19) to that random walk and get
m(v) 1 − p(v, v − ) F (v, w) ETv (Fv , Fw ) = . GTv (v, v) We now apply (10.8) and (10.9), recalling in addition that pTv (v, u) =
p(v, u) 1 − p(v, v − )
for u ∈ Tv , and obtain 1 − p(v, v − ) GTv (v, v)
= 1 − p(v, v − ) − 1 − p(v, v − ) UTv (v, v) X = 1 − p(v, v − ) − p(v, u)F (u, v) u:u− =v
= 1 − p(v, v ) − U (v, v) − p(v, v − )F (v − , v) ν v (∂Tv ) 1 − p(v, v − ) 1 − F (v − , v) = , = G(v, v) G(v, v) −
where in the last step we have used Lemma 10.6. We have obtained ETv (Fv , Fw ) =
m(v)F (v, w) ν v (∂Tv ). G(v, v)
In the same way, exchanging roles between Tw and Tv and using reversibility (10.13), ETw (Fv , Fw ) =
m(w)F (w, v) m(v)F (v, w) m(v) ν w (∂Tw ) = ν w (∂Tw ) = ν v (∂Tw ) G(w, w) G(v, v) G(v, v)
Putting things together, we get that ET (hϕ , hψ ) = ETw (Fv , Fw )ν v (∂T \ ∂Tv ) = 66
m(v) ν v (∂Tw )ν v (∂T \ ∂Tv ), G(v, v)
as proposed. Case 2. Tw ∩ Tv = ∅ . In view of Lemma 10.5, both sides of the D-N-formula are independent of the root o. Thus we may declare our root to be one of the neighbors of v that is not on π(v, w). Also, let v¯ be the neighbour of v on π(w, v). Then, with our chosen new root, the complement of the “old” Tv is Tv¯ , which contains Tw (The latter remains the same with respect to the new root). Thus, we can apply the result of case 1 to Tv¯ and Tw . This means that we have to replace the functions ϕ and hϕ with 1 − ϕ and 1 − hϕ , respectively, which just means that we change the sign on both sides of the identity. We are re-conducted to Case 1 without further computations. We deduce from what we have done so far, and from linearity of the Poisson transform as well of bilinearity of the forms on both sides of the D-N-formula , that it holds for linear combinations of indicator functions of sets ∂Tv . Those indicator functions are dense in the space C(∂T ) with respect to the max-norm. Thus, the D-N-formula holds for all continuous functions on ∂T . The extension to all of D(∂T, P) is by standard approximation.
11
The duality of random walks on trees and isotropic processes on their boundaries
When looking at our isotropic processes and at the boundary process of Kigami [34], it is natural to ask the following two questions. Question I. Given a transient random walk on T associated with the Dirichlet form ET of (10.14), does the boundary process on ∂T induced by the form EHD of (10.16) arise as one of the isotropic processes (1.4) on ∂T with transition probabilities (10.6), with respect to the measure µ = ν o on ∂T , some ultra-metric element φ on T and a suitable distance distribution σ on [0 , ∞) ? Question II. Conversely, given data µ, φ and σ, is there a random walk on T with limit distribution ν o = µ such that the isotropic process induced by µ, φ and σ is the boundary process with Dirichlet form EHD ? Before answering both questions, we need to specify the assumptions more precisely. When starting with (φ, µ, σ), we assume as in §1 that µ is supported by the whole of ∂T . Thus, on the side of the random walk, we also want that supp(ν o ) = ∂T . This is equivalent with the requirement that ν o (∂Tv ) > 0 for every v ∈ T . By (10.11) this is in turn equivalent with F (v, v − ) < 1 for every v ∈ T \ {o}. (11.1) Indeed, we shall see that we need a bit more, namely that lim G(v, o) = 0 ,
v→∞
(11.2)
that is, for every ε > 0 there is a finite set A ⊂ T such that G(v, o) < ε for all v ∈ T \ A. This condition is necessary and sufficient for solvability of the Dirichlet problem: for any ϕ ∈ C(∂X), its Poisson transform hϕ provides the unique continuous extension of ϕ to Tb which is harmonic in T . See e.g. [60, Corollary 9.44]. We shall restrict attention to random walks with properties (11.1) and (11.2) on a rooted tree with forward degrees ≥ 2.
67
Answer to Question I. We start with a random walk that fulfills the above requirements. We know from §1 that each (µ, φ, σ)-process arises as the standard process of Definition 1.4 with respect to the intrinsic metric, see the lines before (1.6): given φ and σ, the intrinsic metric is induced by the ultra-metric element φ∗ (u) = −1 log σ φ(u) . (11.3)
Thus, we can eliminate σ from our considerations by just looking for an ultra-metric element φ such that the boundary process is the standard process on ∂T associated with (ν 0 , φ). Since the processes are determined by the Dirichlet forms, we infer from Theorems 7.2 and 10.4 that we are looking for φ such that J(x, y) = Θ(x, y) for all x, y ∈ ∂T with x 6= y, where J(x, y) is given by (7.1). Rewriting J(x, y) in terms of φ, ν o and the tree structure, this becomes Z 1/φ(x∧y) m(o) dt 1 + . (11.4) = φ φ(o) G(o, o)F (o, x ∧ y)F (x ∧ y, o) ν o B1/t (x) 1/φ(o) In our case, m(o) = 1, but we keep track of what happens when one changes the root or the normalisation of m. First of all, since deg+ (o) ≥ 2, there are x, y ∈ ∂T such that x ∧ y = o. We insert these two boundary points in (11.4). Since F (o, o) = 1, we see that we must have φ(o) = G(o, o)/m(o) . Now take v ∈ T \ {o}. Since forward degrees are ≥ 2, there are x, y, y ′ ∈ ∂T such that x ∧ y = v and x ∧ y ′ = v − . We write (11.4) first for (x, y ′ ) and then for (x, y) and then take the difference, leading to the equation Z 1/φ(v) m(o) m(o) dt − . (11.5) = φ G(o, o)F (o, v)F (v, o) G(o, o)F (o, v − )F (v − , o) (x) 1/φ(v− ) ν o B 1/t
φ (v) = ∂Tv , whence that integral By (10.5), within the range of the last integral we must have B1/t reduces to 1 1 1 − φ(v) φ(v − ) ν o (∂Tv )
We multiply equation (11.5) by ν o (∂Tv ) and simplify the resulting right hand side m(o) m(o) − ν o (∂Tv ) G(o, o)F (o, v)F (v, o) G(o, o)F (o, v − )F (v − , o)
by use of the identities (10.7) – (10.10)) and the first of the two formulas of (10.11) (for ν o ). We obtain that the ultra-metric element that we are looking for should satisfy 1 m(o) m(o) 1 − = − φ(v) φ(v − ) G(v, o) G(v − , o)
for every v ∈ T \ {o} .
(11.6)
This determines 1/φ(v) recursively, and with m(o) = 1, we get φ(v) = G(v, o) . Since by (10.7) and (10.10) G(v, o) = F (v, v − )G(v − o), the assumptions (11.1) and (11.2) yield that φ is an ultra-metric element. Tracing back the last computations, we find that with this choice of φ, we have indeed that J(x, y) = Θ(x, y) for all x, y ∈ ∂T with x 6= y. We have proved the following. 68
Theorem 11.1 Let T be a locally finite, rooted tree with forward degrees ≥ 2. Consider a transient nearest neighbour random walk on T that satisfies (10.7) and (10.10). Then the boundary process on ∂T induced by the Dirichlet form (10.16) coincides with the standard process associated with ultra-metric element φ = G(·, o) and the limit distribution ν o of the random walk. Let L be the Laplacian associated with the boundary process of Theorem 11.1. L acts on locally constant functions f by Z f (x) − f (y) Θo (x, y) dν o (y). Lf (x) = ∂T
In view of the identification of balls in ∂T with vertices of T, the functions of (7.6) now become fv =
1∂Tv− 1∂Tv , v ∈ T \{o}. − ν o (∂Tv ) ν o (∂Tv− )
In addition, we set fo = 1 and note that it is an eigenfunction of L with eigenvalue 0. Applying Theorem 7.3 we obtain Corollary 11.2 For v ∈ T \{o}, we have Lfv = G(v − , o)−1 fv . In particular, Spec(L) = {G(v, o)−1 : v ∈ T } ∪ {0}. Remark 11.3 For any two vertices v and w in T \{o} such that v − = w− = u the functions fv and fw are eigenfunctions of L corresponding to the eigenvalue λ = 1/G(u, o). Hence the eigenspace H(u) corresponding to the vertex u is spanned by functions {fv : v − = u}. Since the rank of the system {fv : v − = u} is deg+ (u) − 1, where deg+ (u) ≥ 2 is the forward degree of the vertex u, we obtain dim H(u) = deg+ (u) − 1. Remark 11.4 Given the random walk on T and the associated boundary process on ∂T , we might want to realize it as the (ν o , φ, σ)-process for an ultra-metric element φ different from G(·, o). This means that we have to look for a suitable distance distribution σ on [0 , ∞), different from the inverse exponential distribution (1.5). In view of (11.3), we are looking for σ such that for our given generic φ, σ φ(v) = e−1/G(v,o) .
For this it is necessary that φ(u) = φ(v) whenever G(u, o) = G(v, o): we need φ to be constant on equipotential sets. In that case, the distribution function σ(r) is determined by the above equation for r in the value set Λφ of the ultra-metric dφ . We can “interpolate” that function in an arbitrary way (monotone increasing, left continuous) and get a feasible measure σ.
Answer to Question II. Answering question (II) means that we start with φ and µ and then look for a random walk with limit distribution ν o = µ such that the standard (φ, µ)-process is the boundary process associated with the random walk. We know from Theorem 11.1 that in this case, we should have φ(v) = G(v, o), whence in particular, φ(o) > 1. Thus we cannot expect that every φ is suitable. The most natural choice is to replace φ by C · φ for some constant C > 0. For the standard processes associated with φ and C · φ, respectively, this just gives rise of a linear time change: if the old process is {Xt }t>0 , then the new one is {Xt/C }t>0 . 69
Theorem 11.5 Let T be a locally finite, rooted tree with forward degrees ≥ 2. Consider an ultra-metric element φ on T and a fully supported probability measure µ on ∂T . Then there are a unique constant C > 0 and a unique transient nearest neighbour random walk on T that satisfies (10.7) and (10.10) with the following properties: 1. µ = ν o is the limit distribution of the random walk. 2. The associated boundary process coincides with the standard process on ∂T induced by the ultra-metric element C · φ and the given measure µ. For the proof, we shall need three more formulas. The first two are taken from [60, Lemma 9.35], while the third is immediate from (10.11) and (10.10) F (u, v) if u ∼ v , 1 − F (u, v)F (v, u) X F (u, v)F (v, u) G(u, u) = 1 + 1 − F (u, v)F (v, u) v:v∼u
G(u, u) p(u, v) =
F (v − , v) =
and
ν o (∂Tv )/F (o, v − ) . 1 − F (v, v − ) + F (v, v − ) ν o (∂Tv )/F (o, v − )
(11.7) (11.8) (11.9)
Proof of Theorem 11.5. We proceed as follows: we start with φ and µ and replace φ by a new ultra-metric element C · φ, with C to be determined, and µ being the candidate for the limit distribution of the random walk that we are looking for. Using the various formulas at our disposal, we first construct in the only possible way the quantities F (u, v) , u, v, ∈ T , in particular when u ∼ v. In turn, they lead to the Green kernel G(u, v). So far, these will be only “would-be” quantities whose feasibility will have to be verified. e v). Via (11.7), they will lead to Until that verification, we shall denote them Fe(u, v) and G(u, definitions of transition probabilities p(u, v). Stochasticity of the resulting transition matrix P will also have to be verified. e v) really is the Green Only then, we will use a potential theoretic argument to show that G(u, kernel associated with P, so that the question mark that is implicit in the “ e ” symbol can be removed. First of all, in view of Theorem 11.1, we must have
whence by (10.7) and (10.10)
and more generally
e o) , C · φ(v) = G(v,
Fe(v, v − ) = φ(v)/φ(v − )
for v ∈ T \ {o} ,
Fe(v, u) = φ(v)/φ(u) when u ≤ v .
(11.10)
We note immediately that 0 < Fe(v, u) < 1 when u < v, and that Fe(u, u) = 1. Next, we use (11.9) to construct recursively Fe(v − , v) and Fe(o, v). We start with Fe(o, o) = 1. If v 6= o and Fe(o, v − ) is already given, with µ(∂Tv− ) ≤ Fe(o, v − ) ≤ 1 70
(the lower bound is required by (10.11)), then we have to set
and
Fe(v − , v) =
µ(∂Tv )/Fe(o, v − ) 1 − Fe(v, v − ) + Fe(v, v − ) µ(∂Tv )/Fe(o, v − ) Fe(o, v) = Fe(o, v − )Fe (v − , v) .
Since we see that
(11.11)
Fe(o, v − ) ≥ µ(∂Tv− ) ≥ µ(∂Tv ), 0 < Fe(v − , v) ≤ 1.
We set – as imposed by (10.10) –
Fe (o, v) = Fe(o, v − )Fe(v − , v).
Formula (11.11) (re-)transforms into
µ(∂Tv ) = Fe (o, v − )Fe(v − , v)
1 − Fe (v, v − ) ≤ Fe (o, v) ≤ 1 , 1 − Fe (v, v − )Fe (v − , v)
(11.12)
as needed for our recursive construction. At this point, we have all Fe (u, v), initially for u ∼ v, and consequently for all u, v by taking products along geodesic paths. We now can compute the constant C: (11.8), combined with (11.10) and (11.12) for u ∼ o forces e o) = 1 + Cφ(o) = G(o, = 1+
X
u:u∼o
= 1+ Therefore C=
X
X
u:u∼o
Fe(o, u)Fe (u, o) 1 − Fe (o, u)Fe(u, o)
Fe(u, o) µ(∂Tu ) 1 − Fe (u, o)
φ(u)/φ(o) µ(∂Tu ) 1 − φ(u)/φ(o) u:u∼o
X φ(u)/φ(o) 1 + µ(∂Tu ) . φ(o) u:u∼o φ(o) − φ(u)
e u) via (11.8): We now construct G(u,
e u) = 1 + G(u,
X
v:v∼u
Fe(u, v)Fe (v, u) . 1 − Fe(u, v)Fe(v, u)
(11.13)
(11.14)
For u = o, we know that this is compatible with our choice of C. At last, our only choice for the Green kernel is e v) = Fe (u, v)G(v, e v) , u, v ∈ T . G(u,
Now we finally arrive at the only way how to define the transition probabilities, via (11.7): p(u, v) =
Fe (u, v) . e u) 1 − Fe(u, v)Fe (v, u) G(u, 1
71
(11.15)
Claim 1.
P is stochastic.
Proof of Claim 1. Combining (11.15) with (11.14), we deduce that we have to verify that for every u ∈ T , X Fe(u, v) 1 − Fe(v, u) = 1. (11.16) 1 − Fe (u, v)Fe(v, u) v:v∼u
If u = o, then by (11.12) this is just
X
µ(∂Tv ) = 1.
v:v∼o
If u 6= o then, again by (11.12), the left hand side of (11.16) is X Fe(u, v) 1 − Fe(v, u) Fe(u, u− ) 1 − Fe(u− , u) + 1 − Fe(u, v)Fe(v, u) 1 − Fe (u, u− )Fe(u− , u) − v:v =u
=
This proves Claim 1.
X µ(∂Tv ) 1 − Fe (u− , u) = 1. +1− e(u, u− )Fe(u− , u) e (o, u) 1 − F F − v:v =u
e u0 ) satisfies P g˜u = g˜u − 1u . Claim 2. For any u0 ∈ T , the function g˜u0 (u) = G(u, 0 0 0
Proof of Claim 2. First, we combine (11.14) with (11.15) to get P g˜u0 (u0 ) =
X
v:v∼u0
e 0 , u0 ) = p(u0 , v)Fe (v, u0 )G(u
X
v:v∼u0
Fe(u0 , v)Fe(v, u0 ) = g˜u0 (u0 ) − 1 , 1 − Fe (u0 , v)Fe (v, u0 )
and Claim 2 is true at u = u0 . Second, for u 6= u0 , let w be the neighbour of u on π(u, u0 ). Then X e u0 ) + p(u, w)G(w, e P g˜u0 (u) = p(u, v)Fe(v, u)G(u, u0 ) v:v∼u,v6=w
=
X
e u0 ) Fe(u, v)Fe(v, u) G(u, e u0 ) + p(u, w)G(w, e − p(u, w)Fe(w, u)G(u, u0 ) e(u, v)Fe (v, u) G(u, e u) 1 − F v:v∼u | {z } e G(u,u)−1
= G(u, u0 ) 1 − since
1
e u) G(u,
− p(u, w)Fe (w, u) + p(u, w)
1
Fe(u, w)
!
= g˜u0 (u)
e u) p(u, w)/Fe(u, w) − p(u, w)Fe(w, u) = 1/G(u,
by (11.15). This completes the proof of Claim 2.
Now we can conclude: the function g˜u0 is non-constant, positive and superharmonic. Therefore the random walk with transition matrix P given by 11.15 is transient and does posses a Green function G(u, v). Furthermore, by the Riesz decomposition theorem, we have g˜u0 = Gf + h , where h is a non-negative harmonic function and the charge f of the potential X Gf (u) = G(u, v)f (v) v
72
is f = g˜u0 − P g˜u0 = 1u0 . That is, e u0 ) = G(u, u0 ) + h(x) G(u,
for all u ∈ T.
Now let x ∈ ∂T and v = u0 ∧ x. If u ∈ Tv then by our construction e e u0 ) = G(u, e o) G(v, u0 ) φ(u) → 0 G(u, φ(v)
as u → x .
e u0 ) vanishes at infinity, and the same must hold for h. By the Maximum Principle, Therefore G(·, h ≡ 0. e v) = G(u, v) for all u, v ∈ T . But then, by our construction, also We conclude that G(u, Fe(u, v) = F (u, v), the “first hitting” kernel associated with P. Comparing (11.12) with (10.11), we see that µ = ν o . This completes the proof. The non-compact case. Our general approach in the present work is not restricted to compact spaces. In case of a non-compact, locally compact ultra-metric space without isolated points, one constructs the tree in the same way: the vertex set corresponds to the collection of all closed balls, and neighbourhood in the resulting tree is defined as above: if a vertex v corresponds to a ball B, then the predecessor v − is the vertex corresponding to the ball B ′ (see Definition 2.1), and there is the edge [v − , v]. Now every vertex has a predecessor (while in the compact case, the root vertex has none), and the tree has its root at infinity, i.e., the ultra-metric space becomes ∂ ∗ T = ∂T \ {̟}, where ̟ is a fixed reference end of T . See Figure 5 below. We now start with this situation: given a tree T and a reference end ̟ ∈ ∂T , the predecessor − − of a vertex v with respect to ̟ is the neighbour of v on the geodesic π(v, ̟). Given v = v̟ two elements w, z ∈ Tb \ {̟}, their confluent w f z with respect to ̟ is again defined as the last common element on the geodesics π(̟, w) and π(̟, z), a vertex, unless v = w ∈ ∂ ∗ T (Figure 5). Again, it is natural to assume that each vertex has at least two forward neighbours. In this situation, for the Definition 10.1 of an ultra-metric element φ : T → (0 , ∞), we need besides monotonicity [φ(v) < φ(v − )] that φ tends to ∞ along π(o, ̟), while it has to tend to 0 along any geodesic going to ∂ ∗ T . The associated ultra-metric on ∂ ∗ T is then given in the same way as before: ( 0, if x = y , dφ (x, y) = φ(x f y) , if x 6= y . Let us note here that also when φ does not tend to ∞ along π(x, ̟), this does define an ultrametric, but then (∂ ∗ T, dφ ) will not be complete. Also, if the inequality φ(v) ≤ φ(v − ) is not strict, one gets an ultra-metric, but then the above construction of the tree of closed balls does not recover the original tree from (∂ ∗ T, dφ ). Finally, if φ does not tend to 0 along some geodesic π(o, x), x ∈ ∂ ∗ T , then x will be an isolated point in (∂ ∗ T, dφ ). (The last two observations are also true in the compact case, for a tree with a root vertex.) Returning to our setting, the reference measure µ of a (φ, µ, σ)-process may have infinite mass: a Radon measure supported on the whole of ∂ ∗ T . Again, we know that it is sufficient to study the standard (φ, µ)-process. We give a brief outline of the duality of such processes with random walks on T . This should be compared with the final part of Kigami’s second paper [35] (whose preprint became available when the largest part of this work had been done, and 73
in particular, the preliminary version [61] of the present random walk sections had first been circulated.). With respect to ̟, the branch of T rooted at u ∈ T is now Tu = T̟,u = {v ∈ T : u ∈ π(v, ̟)}. Then ∂Tu is a compact subset of ∂ ∗ T , a ball with dφ -diameter φ(u). Here, it will be good to write To,u for the branch with respect to a root vertex o ∈ T , as defined in (10.2). We note that T̟,u = To,u iff u ∈ / π(o, ̟). In addition to the reference end ̟, we choose such a root o and write on for its n-th predecessor, that is, the vertex on π(o, ̟) at graph distance n from o. ̟ .. . H−2
H−1 H0 H1 .. .
. ............. . ... . . . ... .. ... 2 ........ ..•... ... ... ... ..... ... ... . . . ... ... ... ... ... .. ... . . ... .. . . ... .. . ... . ... ... . ... . . . ... .. . ... . .. ... . . ... .. . . ... 1 .... ..... • .... . ..• . .. .... ... ..... . . . . ... ... . .. . . . . . ... . ... . ... ... ... ... ... ... .. ... ... ... ... ... . . . . ... ... .. .. . . ... . . . ... ... ..... . . . .... • . . . . .• . .. .... . ... ..... ... ..... ... .. . . . . .. ..... . .. .... . . . . . . . . . . . . ... ... ... ... .. .. .. .. . . . . ... . . . . . . . .. .. .. .. . .. ...... ....... ....... ....... ....... ....... ....... ... .... ..• . ... .. ... .. ... .. ... .. ... .. ... .. ... .. ... ..
̟.
o
o
ufv v
o
u
.. .
.. .
...... ........ ... ... ... ... ... ... ... ... ...... ... ..... .. ... . ... ... ... . . ... .. ... ... ... . . ... ... ... ... . . . .... ... ... ... ..... ... ... . . . ... ... ... .. ... ... ... . . ... .. . . . ..... . ...... . ... .. ... ..... . .. ..... . . . . . . . ... ... .. .. . . ... . . . .. .. .. ....... ....... ....... ....... ... .. ... .. ... .. ... ..
...
...
∂∗T Figure 5 Now let P = p(u, v) u,v∈T be the transition matrix of a transient nearest neighbour random walk on T . We assume once more that (11.1) holds: F (v, v − ) < 1 for every v ∈ T , but now predecessors refer to ̟. (Indeed, this implies (11.1) with respect to any choice of the root vertex.) We now consider the Dirichlet form EHD and look at the formula of Theorem 10.4. We would like to move o to ̟ in that formula. We know from Lemma 10.5 that the measures Θon (x, y) dν on (x) dν on (y) are the same for all n. However, the measures ν on restricted to ∂ ∗ T will typically converge vaguely to 0. Thus, we normalise by defining 2 1 µn = ν on and Jn (x, y) = Θon (x, y) ν on (∂To ) . ν on (∂To ) For the following, recall that Tu = Tu̟ , and note that u f o = ok for some k ≥ 0.
Lemma 11.6 Let A ⊂ ∂ ∗ T be compact, so that there is a vertex u such that A ⊂ ∂Tu . If u f o = ok then for all n ≥ k and for all x, y ∈ ∂Tok , µn (A) = µk (A) =: µ(A) ,
and Jn (x, y) = Jk (x, y) =: J(x, y).
We have J(x, y) = j(x f y)
with
j(v) =
ϑ2 G(v, v) , v∈T, 2 K(v, ̟) m(v)
where
m(o)ν o (∂To ) , G(o, o) is the Martin kernel at ̟. ϑ=
and K(v, ̟) =
74
F (v, v f o) F (v, v ∧o ̟) = F (o, v f o) F (o, v fo ̟)
Proof. Since ∂Tok contains both ∂To and A, we have for n ≥ k µn (A) =
F (on , ok )ν ok (A) ν on (A) = µk (A). = ν on (∂To ) F (on , ok )ν ok (∂To )
Analogously, Let x, y ∈ ∂Tok and xfy = v, an element of Tok . We use the identity m(v)G(v, w) = m(w)G(w, v), which implies m(on )F (on , o) = m(on )G(on , o)/G(o, o) = m(o)G(o, on )/G(o, o), and compute for n ≥ k Jn (x, y) = =
m(on )2 F (on , o)2 ν on (∂To )2 m(on ) = ν o (∂To )2 F (on , v)G(v, on ) m(on )F (on , v)G(v, on ) 2 2 2 ν o (∂To ) m(o) G(o, on ) G(v, v) , G(o, o)2 G(v, on )2 m(v)
which yields the proposed formula, since G(o, on ) = F (o, o f v)G(o f v, on ) and G(v, on ) = F (v, o f v)G(o f v, on ). Now its is not hard to deduce the following. Theorem 11.7 Let T and its reference end ̟ be as outlined above. Consider a nearest neighbour random walk on T that satisfies F (v, v − ) < 1 for every v ∈ T . Let µ and J be as in Lemma 11.6. Then for all compactly supported continuous functions ϕ, ψ on ∂ ∗ T , the Dirichlet form (10.16) can be written as Z ϕ(x)ψ(x) dµ(x) , where EHD (ϕ, ψ) = EJ (ϕ, ψ) + ϑ · ν o ({̟}) ∂∗T Z Z 1 EJ (ϕ, ψ) = ϕ(x) − ϕ(y) ψ(x) − ψ(y) J(x, y) dµ(x) dµ(y) . 2 ∂∗T ∂∗T
When the random walk is Dirichlet regular (in which case ν o ({̟}) = 0), the form EJ = EHD induces the standard (µ, φ)-process, where the ultra-metric element φ with respect to ̟ is given by 1 φ(v) = K(v, ̟) , ϑ and ϑ and the Martin kernel K(v, ̟) are as defined in Lemma 11.6. In particular, the (µ, φ)-process is the boundary process with a time-change. Proof. There is k such that the compact supports of ϕ and ψ are contained in ∂Tok . Let n ≥ k. Using lemmas 10.5 and 11.6, Z Z 1 EHD (ϕ, ψ) = ϕ(x) − ϕ(y) ϕ(x) − ϕ(y) Jn (x, y) dµn (x) dµn (y) 2 ∂T ∂T Z Z 1 ϕ(x) − ϕ(y) ϕ(x) − ϕ(y) J(x, y) dµ(x) dµ(y) = 2 ∂Ton ∂Ton Z Z Jn (x, y) dµn (y) dµ(x) ϕ(x)ψ(x) + ∂T \∂Ton ∂Ton | {z } =: fn (x)
As n → ∞,
1 2
Z
∂Ton
Z
∂Ton
ϕ(x) − ϕ(y) ϕ(x) − ϕ(y) J(x, y) dµ(x) dµ(y) → EJ (ϕ, ψ) . 75
Let us look at the second term. We have Z Θon (x, y) ν on (∂To ) dν on (y). fn (x) = ∂T \∂Ton
For x ∈ ∂Ton and y ∈ ∂T \ ∂Ton , their confluent with respect to on is on itself. Therefore, using (10.17) and (10.11) Θon (x, y) ν on (∂To ) = =
m(on )G(on , o) m(on ) F (on , o) ν o (∂To ) = ν o (∂To ) G(on , on ) G(on , on )G(o, o) m(o)G(o, on ) m(o) ν o (∂To ) = ν o (∂To ) F (o, on ) = ϑ F (o, on ) . G(on , on )G(o, o) G(o, o)
Now note that for y ∈ ∂T \ ∂Ton , we have F (o, on ) dν on (y) = dν o (y). Therefore Z F (o, on ) dν on (y) = ϑ · ν o (∂T \ ∂Ton ) → ϑ · ν o ({̟}) , fn (x) = ϑ ∂T \∂Ton
and as n → ∞ , we can use dominated convergence to get that Z Z Z ϕ(x)ψ(x) fn (x) dµ(x) Jn (x, y) dµn (y) dµ(x) = ϕ(x)ψ(x) ∂∗T ∂T \∂Ton ∂Ton Z ϕ(x)ψ(x) dµ(x) , → ϑ · ν o ({̟}) ∂∗T
as proposed. To prove the formula for the associated ultra-metric element, we proceed as in the proof of Theorem 11.1, see (11.4) and the subsequent lines. We find that the ultra-metric element must satisfy 1 1 − − = j(v) − j(v ) µ(∂Tv ) . φ(v) φ(v − )
The right hand side of this equation can be computed: we have v − f o = ok for some k ≥ 0, and combining the arguments after (11.4) with those of the proof of Lemma 11.6, m(ok ) m(ok ) − − ν ok (∂Tv ) ν ok (∂To ) j(v) − j(v ) µ(∂Tv ) = F (ok , v)G(v, ok ) F (ok , v − )G(v − , ok ) m(ok ) m(ok ) = − F (ok , o) ν o (∂To ) G(v, ok ) G(v − , ok ) G(o, ok ) G(o, ok ) m(o)ν o (∂To ) ϑ ϑ = − = − G(v, ok ) G(v − , ok ) G(o, o) K(v, ̟) K(v − , ̟)
We infer that 1/φ(·) − ϑ/K(·, ̟) must be constant. By Dirichlet regularity of the random walk, K(v, ̟) → ∞ as v → ̟. On the other hand, also φ(on ) must tend to infinity. Thus, the constant is 0, and φ has the proposed form. Lemma 11.6 and Theorem 11.7 lead to clearer insight and simpler proofs concerning the material on random walks in [35, §10 – §11], in particular [35, Theorem 11.3]. Namely, our limit measure µ coincides with the ν ∗ of [35]. We also note here, that there are examples where µ(∂ ∗ T ) = ∞, as well as examples where µ(∂ ∗ T ) < ∞, even though the ultra-metric space is non-compact. In the present work, we have always assumed that the reference measure has infinite mass in the non-compact case, but this is not crucial for our approach.
76
Remark 11.8 In the present sections 10 – 11, we have always assumed that the ultra-metric space has no isolated points, which for the tree means that deg+ ≥ 2. Theme of [7] is the opposite situation, where all points are isolated, i.e., the space is discrete. In that case the ultra-metric space is also the boundary of a tree, which does not consist of ends, but of terminal vertices, that is, vertices with only one neighbour. From the point of view of the present section, the mixed situation works equally well. If we start with a locally compact ultra-metric space having both isolated and non-isolated points, we can construct the tree in the same way. The vertex set is the collection of all closed balls. The isolated points will then become terminal vertices of the tree, which have no neighbour besides the predecessor, as for example the vertices x and y in Figure 6. All interior (non terminal) vertices will have forward degree ≥ 2. ̟ . ............. ... ... . . ... ... ... ... . . . ....... ... ..... ... ... ... ... . . ... ... ... .. . ... . .. ... . . ... .. . . ... .. . ... . ... ... . . ... . . ... .. . . ... .. . ... . ... ... . ... . . . • ...... . . ... ..... . . ... . . ... .. . . ... .. ... ... ... ... ... ... . ..... . • . ...... .. .... . ... ... . .. .... ... ..... . . . . ... . . . .... . . ... • ............. ... ... ... ... ... ... ... ... .. ..
y
o
x
.. .
̟.
...... ........ ... ... ... ... ... ... ... .. ...... ... ... .. ..... . . .. ... ... .. ... . . .. ... ... ... .. . . .. ... ... ... .. . . ...... ... ... ... ..... ... ... ... ... . . ... ... ... ... ... ... ... . .... ... . . ... ... .. .... . . ... ..... ... . .. . . . . ... ... . . . .... . . . ... .... . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
...
...
.. .
Figure 6 In the compact case, the boundary ∂T of that tree consists of the terminal vertices together with the space of ends. In the non-compact case, we will again have a reference end ̟ as above, and ∂ ∗ T consists of all ends except ̟, plus the terminal vertices. The definition of an ultra-metric element remains the same, but we only need to define it on interior vertices. In this general setting, the construction of (φ, µ, σ)-processes remains unchanged. Even in presence of isolated points, the duality between (φ, µ, σ)-processes and random walks on the associated tree remains as explained here. The random walk should then be such that the terminal vertices are absorbing, and that the Green kernel tends to 0 at infinity. The Doob-Na¨ım formula extends readily to that setting. Remark 11.9 Let us again consider the general situation when we start with a transient random walk on a locally finite, rooted tree T . The limit distribution ν o will in general not be supported by the whole of ∂T . The boundary process can of course still be constructed, see [34], but will evolve naturally on supp(ν o ) only. Thus, we can consider our ultra-metric space to be just supp(ν o ). The tree associated with this ultra-metric space will in general not be the tree we started with, nor its transient skeleton as defined in [60, (9.27)] (the subtree induced by o and all v ∈ T \ {o} with F (v, v − ) < 1, where v − = vo− ). The reasons are twofold. First, the construction of the tree associated with supp(ν o ) will never give back vertices with forward degree 1. Second, some end contained in supp(ν o ) may 77
be isolated within that set, while not being isolated in ∂T . But then this element will become a terminal vertex in the tree associated with the ultra-metric (sub)space supp(ν o ). This occurs precisely when the transient skeleton has isolated ends. Thus, one should work with a modified “reduced” tree plus random walk in order to maintain the duality between random walks and isotropic jump processes. The same observations apply to the non-compact case, with a reference end in the place of the root and the measure µ of Lemma 11.6 in the place of ν o . Remark 11.10 Given a transient random walk on the rooted tree T , [34] also recovers an intrinsic metric of the boundary process on ∂T (compact case !) in terms of what is called an ultra-metric element in the present paper. This is of course φ(x) = G(x, o), denoted Dx in [34], where it is shown that for ν o -almost every ξ ∈ ∂T , Dx → 0 along the geodesic ray π(o, ξ). This has the following potential theoretic interpretation. A point x ∈ ∂T is called regular for the Dirichlet problem, if for every ϕ ∈ C(∂T ), its Poisson transform hϕ satisfies lim hϕ (v) = ϕ(x). v→x
It is known from Cartwright, Soardi and Woess [17, Remark 2] that x is regular if and only if limu→x G(u, o) = 0 (as long as T has at least 2 ends), see also [60, Theorem 9.43]. By the latter theorem, the set of regular points has ν o -measure 1. That is, the Green kernel vanishes at ν o -almost every boundary point. Remark 11.11 In the proof of Theorem 11.5, we have reconstructed random walk transition probabilities from C · φ(u) = G(u, o) and µ = ν o . ˇek [57]: how to reconstruct A similar (a bit simpler) question was addressed by Vondrac the transition probabilities from all limit distributions ν u , u ∈ T , on the boundary. This, as well as our method, basically come from (10.11) and (11.7) + (11.8), which can be traced back to Cartier [15].
12
Random walk associated with p-adic fractional derivative
In conclusion we consider a two-fold specific example which unites the approaches of Sections 8–9 and Sections 10–11. We start with the compact case. The p-adic fractional derivative on Zp . Let Zp ⊂ Qp be the group of p-adic integers. As a counterpart of the operator Dα we introduce the operator Dα of fractional derivative on Zp . We show that it is the Laplacian of an appropriate isotropic Markov semigroup. Then we construct a random walk associated with Dα in the sense of Sections 10–11 . cp is a discrete Abelian group. It is known Since Zp is a compact Abelian group, its dual Z c that the group Zp can be identified with the group Z(p∞ ) = {p−n m : 0 ≤ m < pn , n = 1, 2, ...}
equipped with addition of numbers mod 1 as the group operation. As sets (but not as groups) Z(p∞ ) ⊂ Qp , whence the function ξ 7→ kξkp is well-defined on the group Z(p∞ ).
78
Definition 12.1 The operator (Dα , D), α > 0, is defined via the Fourier transform on the compact Abelian group Zp by α f (ξ) = kξkα fb(ξ), ξ ∈ Z(p∞ ) , d D p
where D is the space of locally constant functions on Zp (Compare with the Definition 8.1 of the operator Dα .)
An immediate consequence is that the operator Dα is a non-negative definite self-adjoint operator whose spectrum coincides with the range of the function ξ 7→ kξkαp : Z(p∞ ) → R+ , that is, Spec(Dα ) = {0, pα , p2α , ...}. The eigenspace H(λ) of the operator Dα corresponding to the eigenvalue λ = pkα , k ≥ 1, is spanned by the function fk =
1 1 1k − 1 k−1 µp (pk Zp ) p Zp µp (pk−1 Zp ) p Zp
and its shifts fk (· + a) with any a ∈ Zp /pk Zp . Indeed, computing the Fourier transform of the function fk ,
we obtain
fbk (ξ) = 1{kξkp ≤pk } − 1{kξkp ≤pk−1 } = 1{kξkp =pk } , α f (ξ) = kξkα fb (ξ) = pkα fb (ξ). [ D k k p k
The maximal number of linearly independent functions in the set {fk (· + a) : a ∈ Zp /pk Zp } is pk−1 ( p − 1), whence dim H(λ) =pk−1 (p − 1). All the above shows that Dα coincides with the Laplacian (i.e., = −Dα is the Markov generator) of some isotropic Markov semigroup (Pαt )t>0 defined on the ultra-metric measure space (Zp , dp , µp ) as constructed in (1.2)–(1.4) and studied in the previous sections. In particular, using the complete description of the set Spec(Dα ) we compute the intrinsic distance, call it dp,α (x, y), kx − ykp α . dp,α (x, y) = p
It is now straightforward to compute the spectral distribution function Nα (x, τ ) ≡ Nα (τ ) and then the jump-kernel Jα (x, y) ≡ Jα (x − y) of the operator Dα . We claim that ! 1 p−α − p−α−1 pα − 1 + Jα (x, y) = . (12.1) 1 − p−α−1 1 − p−α kx − yk1+α p
Recall for comparison that according to (8.2) the jump-kernel Jα (x, y) of the operator Dα is given by 1 pα − 1 . Jα (x, y) = −α−1 1−p kx − yk1+α p 79
To prove (12.1), we compute Jα (z). Let kzkp = p−l , then dp,α (0, z) = p−(l+1)α and
Jα (z) =
1/dp,α Z (0,z)
Nα (τ )dτ =
0
p(l+1)α Z
Nα (τ )dτ .
0
The function Nα (τ ) is a non-decreasing, left-continuous staircase function having jumps at the points τ k = pkα , k = 1, 2, ..., and taking values at these points Nα (τ k ) = pk−1 , whence Jα (z) = 1 · pα + p(p2α − pα ) + p2 (p3α − p2α ) + ... + pl (p(l+1)α − plα ) pα − 1 l(α+1) 1 − p−1 + p = 1 − p−α−1 1 − p−α−1 ! pα − 1 1 p−α − p−α−1 = + 1 − p−α−1 1 − p−α kzk1+α p as desired. Next, we apply Theorem 7.2 and obtain Z α f (x) − f (y) Jα (x − y) dµp (y) D f (x) =
(12.2)
Zp
and
(Dα f, f ) =
1 2
Z
Zp ×Zp
2 f (x) − f (y) Jα (x − y) dµp (x) dµp (y).
The equations (12.1)–(12.2) and (8.1) now yield the following result.
Corollary 12.2 For any function f defined on Zp ⊂ Qp we set fe = f on Zp and 0, otherwise. Then f ∈ Dom(Dα ) =⇒ fe ∈ Dom(Dα ), Dα f (x) = Dα fe(x) and (Dα f, f ) = Dα fe , fe (12.3) whenever x ∈ Zp , f ∈ Dom(Dα ) and (1, f ) = 0.
Nearest neighbour random walk on the rooted tree Top . As an illustration of Theorem 11.5 we construct a random walk on the rooted tree associated with Zp whose boundary process coincides with the isotropic process driven by the operator C · Dα , where C = p−α (1 − p−α ). The Abelian group Zp can be identified with the boundary of the tree Top with root o where every vertex v has p forward neighbours. In our identification, this is the tree of balls of the ultra-metric space (Zp , dp ) with root o corresponding to the whole of Zp and the ultra-metric dp (x, y) = kx − ykp . See Figure 4 above, where p = 2. We fix a constant c ∈ (0, 1) and consider the nearest neighbour random walk on Top with 1/p if v − = o − − p(v, v ) = 1 − c and p(v , v) = . (12.4) c/p otherwise Using [60, Thm. 1.38 and Prop. 9.3 ] one can compute precisely the Green function G(v, o), the hitting probability F (v, o) and other quantities associated with our random walk. In particular, choosing c = (1 + p−α )−1 , we obtain F (v, o) = p−α|v|
and G(v, o) = 80
p−α|v| , 1 − p−α
(12.5)
where |v| is the graph distance from v to o. We see that the Green function vanishes at infinity, whence the random walk is Dirichlet regular. The transition probabilities are invariant with respect to all automorphisms of the tree. Every such automorphism must fix o and every level of the tree. Let ν = ν o be the limit distribution on ∂Top of the random walk starting at o. Then also ν is invariant under the automorphism group of the tree (whose action extends to the boundary). In particular it is invariant under the action of Zp . Thus, under the identification of ∂Top with Zp , we have that ν = µp , the normalized Haar measure of Zp . We now look at the boundary process induced by our random walk as a jump process on Zp . By Theorem 11.1, the boundary process arises as an isotropic jump process with the reference measure µp . Let L be its Laplacian. By Corollary 11.2, the set Spec(L) coincides with the range of the function v 7→ 1/G(v, o), v ∈ Top , together with {0}. In view of the above formula for G(v, o) this means that Spec(L) = {0, (1 − p−α ), pα (1 − p−α ), p2α (1 − p−α ), . . . }. Remember that Spec(Dα ) = {0, pα , p2α , . . . } =
pα Spec(L). 1 − p−α
Since both Dα and L have the same orthonormal basis of eigenfunctions, we conclude that they are proportional, that is, pα L. (12.6) Dα = 1 − p−α Thus, finally we come to the following conclusion Proposition 12.3 The boundary process {Xt }t>0 associated with the random walk defined in (12.4) with parameter c = (1 + p−α )−1 and the isotropic jump process {Xtα }t>0 driven by the operator Dα are related by the linear time change Xt/C = Xtα , where C = p−α (1 − p−α ). The equation (12.6) implies that the jump kernels Jα (x, y) and Θo (x, y) of the operators Dα and L, respectively, are related by Jα (x, y) =
pα Θo (x, y). 1 − p−α
(12.7)
We now show how to compute the Na¨ım kernel Θo (x, y) =
1 , G(o, o)F (o, v)F (v, o)
where v = x ∧ y,
directly, using the data of (12.5). We do not yet have F (o, v). We shall compute N (v) =
1 . F (o, v)F (v, o)
Since it depends only on the level k of v, we consider an arbitrary geodesic ray [o = v0 , v1 , ...] and set up a linear recursion for N (vk ). Denoting by w1 an arbitrary neighbour of o different from v1 and applying [60, Prop. 9.3(b)] and (12.5), we obtain F (o, v1 ) =
1 (p − 1)p−α 1 p−1 + F (w1 , o)F (o, v1 ) = + F (o, v1 ), p p p p 81
whence F (o, v1 ) =
pα (pα
1 . − p + 1)
Thus, we get the initial values N (v0 ) = 1
and N (v1 ) = pα − p + 1.
Next, for k ≥ 1, we let wk+1 be a forward neighbour of vk different from vk+1 . Applying once again [60, Prop. 9.3(b)] and (12.5), we obtain F (vk , vk+1 ) =
pα (p − 1)pα + F (wk+1 , vk )F (vk , vk+1 ) p(pα + 1) p(pα + 1) 1 F (vk−1 , vk )F (vk , vk+1 ). + α p +1
We insert the value F (wk+1 , vk ) = p−α and divide by F (o, vk+1 ) = F (o, vk )F (vk , vk+1 ) = F (o, vk−1 )F (vk−1 , vk )F (vk , vk+1 ). Then we get 1 1 1 1 pα p−1 1 = + + . F (o, vk ) p(pα + 1) F (o, vk+1 ) p(pα + 1) F (o, vk ) pα + 1 F (o, vk−1 ) Now we multiply both sides with 1/F (vk , o) = pαk and get N (vk ) =
1 p−1 pα N (v ) + N (v ) + N (vk−1 ). k+1 k p(pα + 1) p(pα + 1) pα + 1
This is a homogeneous second order linear recursion with constant coefficients. Its characteristic polynomial has roots 1 and pα+1 . Therefore N (vk ) = A + Bp(α+1)k . Inserting the initial values we easily find the values of A and B. In order to get the Na¨ım kernel, we have to multiply by 1/G(o, o) = 1 − p−α . Thus, we get Θo (x, y) =
1 − p−α (1 − p−α )(p − 1) (1 − p−α )(pα+1 − p) (α+1)k + p = Jα (x, y) , pα+1 − 1 pα+1 − 1 pα
as desired.
The random walk corresponding to Dα on Qp . We can combine the preceding considerations with the material of Lemma 11.6 and Theorem 11.7 concerning the duality with random walks in the non-compact case. It is now easy to understand the random walk corresponding to the fractional derivative on the whole of Qp . The tree associated with Qp is the homogenous tree T = Tp with degree p + 1. We have to choose a reference end ̟. Then we can identify its lower boundary ∂ ∗ Tp with the field of p-adic numbers. With respect to ̟, every vertex v has its predecessor v − and p successors. Every subtree Tv = Tv̟ is isomorphic with the rooted tree Top considered above in the compact case of the p-adic integers. In particular, we choose the root vertex o such that ∂To = Zp . See Figure 5 above, where p = 2. 82
We now define the random walk on Tp as in (12.4), but with predecessors referring to ̟ : p(v, v − ) = 1 − c and p(v − , v) = c/p ,
where c = (1 + p−α )−1 .
(12.8)
For the following quantities, see e.g. [59, pp. 423-424]. For all v ∈ Tp , F (v, v − ) = p−α , F (v − , v) = p−1 , G(v, v) =
1 + p−α 1 − p−α−1
and
ν v (∂Tv ) =
1 − p−α . 1 − p−α−1
This yields that the reference measure µ of the boundary process with respect to ̟, as given by Lemma 11.6, is the standard Haar measure of Qp . We compute ϑ = (1 − p−α )/(1 + p−α ). Furthermore, let us set h(v) = d(v, v f o) − d(o, v f o) (where d is the graph metric). This is the horocycle number of v. That is, the vertices with h(v) = k, k ∈ Z, are the elements in the k-th generation Hk of the tree (see Figure 5), and ∂Tv corresponds to a ball with radius p−k in the standard ultra-metric of Qp . Then K(v, ̟) = pα h(v)
and
m(v) = p(α−1)h(v) .
Putting things together, we get φ(v) =
1 + p−α −α h(v) p 1 − p−α
and j(v) =
(1 − p−α )2 p(α+1) h(v) (1 + p−α )(1 − p−α−1 )
Retranslating this into p-adic notation, we conclude that the intrinsic metric and jump kernel of the boundary process with respect to ̟ are given by dφ (x, y) = J(x, y) =
1 − p−α kx − ykαp and 1 + p−α (1 − p−α )2 1 1 − p−α Jα (x, y) , = (1 + p−α )(1 − p−α−1 ) kx − ykα+1 pα + 1 p
where Jα is the jump kernel associated with Dα . So at last, we get the following. Proposition 12.4 The boundary process {Xt }t>0 with respect to the reference end ̟ associated with the random walk (12.8) on Tp and the isotropic jump process {Xtα }t>0 driven by the operator Dα on Qp are related by the linear time change Xt/C ∗ = Xtα , where C ∗ = (1 − p−α )/(pα + 1). Acknowledgement. This work was begun and finished at Bielefeld University (SFB-701). The authors thank S. Albeverio, J. Bellissard, W. Herfort, A.N. Kochubei, S.A. Molchanov, L. Saloff-Coste, I.V. Volovich and E.I. Zelenov for fruitful discussions and valuable comments.
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A. Bendikov: Institute of Mathematics, Wroclaw University Pl. Grundwaldzki 2/4, 50-384 Wroclaw, Poland email:
[email protected] A. Grigoryan: Department of Mathematics, University of Bielefeld, 33501 Bielefeld, Germany email:
[email protected] Ch. Pittet: CMI, Universit´e d’Aix-Marseille I, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France email:
[email protected] W. Woess: Institut f¨ ur Mathematische Strukturtheorie, Technische Universit¨ at Graz Steyrergasse 30, A-8010 Graz, Austria email:
[email protected]
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