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STRONG FELLER PROPERTIES FOR DISTORTED BROWNIAN MOTION AND APPLICATIONS TO FINITE PARTICLE SYSTEMS WITH SINGULAR INTERACTIONS ¨ SERGIO ALBEVERIO, YURI KONDRATIEV, AND MICHAEL ROCKNER Abstract. We prove strong Feller properties for a class of distorted Brownian motions on d . We also construct a weak solution to the corresponding stochastic differential equation starting from any point in {% 6= 0} and staying in {% 6= 0} before possibly going out of any ball in d . Here % is the Lebesgue density of the symmetrizing measure µ. Our condition on the logarithmic derivative ∇% is that it should be locally in Ld+ε , but only with respect to the % symmetrizing measure µ = % dx, not necessarily Lebesgue measure dx. This allows applications to singular situations. In particular, finite particle systems with two body interactions with infinitely strong repulsion can be treated by our results. Among other things it is shown that they never meet no matter what their starting configuration was. Another application treats diffusions in random media. Dedicated to Len Gross on the occasion of his 65th birthday.

Contents 1. Introduction 2. An elliptic regularity result and its consequences 3. Construction of the semigroup and resolvent of kernels 4. Construction of the associated diffusion process 5. Solution to the stochastic equation 6. Applications to stochastic dynamics 6.1. Stochastic gradient dynamics of N -particle systems 6.2. Diffusions in a random media Acknowledgement References

1 3 6 10 14 16 16 18 19 19

1. Introduction Already in the late seventies and early eighties a number of papers (cf. [AH-KS77], [AFKS81], [Fu82a, Fu82b], [Fu84], [AKS86]) were devoted to the study of symmetric distorted Brownian motion (Xt )t≥0 on d with singular drift, i.e. (Xt )t≥0 is the (weak) solution to the stochastic equation

(1.1)

dXt =

√

2 dWt +

∇% (Xt ) dt , X0 = x (∈ % 1

d

),

¨ SERGIO ALBEVERIO, YURI KONDRATIEV, AND MICHAEL ROCKNER

2

with (Wt )t≥0 = Brownian motion on d and % = Lebesgue density of the symmetrizing measure µ. These papers, as well as their subsequent generalizations to infinite dimensional state spaces (see e.g. [AR89], [AR90], [AKR90], [AR91], [ARZ93]), were motivated in part by Mathematical Physics, and mostly based on techniques from the theory of symmetric Dirichlet forms. However, these lists of references are far from being complete and the reader should also consult the standard reference on symmetric Dirichlet forms [FOT94]. In recent years, the interest in equations of type (1.1) has risen again, since generalizations of distorted Brownian motion to infinite dimensional manifolds, so called “configuration spaces”, have been constructed (see e.g. [Os96], [Yo96], [AKR96a, AKR96b], [AKR98a, AKR98b], [R¨ o98], [MR00], [GKLR01]). and identified as describing exactly infinite particle systems in continuum undergoing very singular interactions, which in turn have been in the focus of study in Statistical Mechanics for many years (see e.g. [La77], [Sp86], [Fr87], [Ta96], [Ol94]). In this paper we are merely interested in the case of finitely many particles. The motivation is the following. Dirichlet form techniques give weak solutions to (1.1) (respectively their infinite dimensional analogues), which can only start from a set of points in d (respectively in the infinite dimensional state space) whose complement is of zero capacity, but maybe is non-empty. In general it is impossible or at least extremely difficult to give an explicit analytic description of the “allowed” starting points. In this paper we, however, shall prove that under suitable integrability conditions on ∇% % with respect to µ = % dx (which still allow applications to the said finite particle systems even if the interactions are as singular as they should be in physically relevant models) (1.1) can be (weakly) solved for any initial condition in {% 6= 0}. These conditions are as follows: For % : d → + we shall assume √ 1,2 ( d , dx), % > 0, dx − a.e. (H1) % ∈ Wloc √ |∇ %| |∇%| d+ε =2 √ ∈ Lloc ( d , µ) for some ε > 0. (H2) % %

s,q Here dx denotes Lebesgue measure on d , W(loc) ( d , dx), s > 0, q ≥ 1 the classiq d cal (local) Sobolev space of order s in L(loc) ( , dx), and µ := % dx. Lq(loc) (µ) := Lq(loc) ( d , µ), q > 0, denote the corresponding real (local) Lp -spaces. Corresponding norms are denoted by k · kLq ( d,µ) , k · kW s,q ( d,dx) etc. (H1) alone already implies that the symmetric positive definite bilinear form Z (1.2) E(u, v) := < ∇u, ∇v > d dµ ; u, v ∈ C0∞ ( d )

d

2

d

is closable on L ( , µ) and that its closure (E, D(E)) is a regular local symmetric Dirichlet form (cf. [Fu84]). By the regularity (E, D(E)) is associated with a Hunt process ˜ := (Ω, ˜ F, ˜ (F˜t )t≥0 , (X ˜ t )t≥0 , ( ˜ x )x∈ d ) on d := Alexandrov compactification of ∆ d ∆ , with continuous sample paths and lifetime ζ. Since (E, D(E)) has no zero order (= killing) part, we may assume that

(1.3)

˜ = {ω = (ω(t))t≥0 ∈ C([0, ∞), Ω

d ∆)

| ω(t) = ∆ ∀ t ≥ ζ}

(cf. [FOT94, Theorem 4.5.3]), and (1.4)

˜ ˜ t ≥ 0. X(t) = ω(t), ω ∈ Ω,

STRONG FELLER PROPERTIES

3

˜ t )t≥0 weakly solves (1.1) under ˜ x for x ∈ d outside a set of capacity zero. We (X refer to [Fu80] and [FOT94] for details and terminology. We shall also discuss the case where in addition to (H1), (H2), we also have that (E, D(E)) is conservative (cf. (H3) in Section 3 below), in which we get slightly more refined results. We note that (as will be explained in the main body of the paper) (H2) implies that % is continuous (or more precisely has a H¨ older-continuous dx-version, cf. Corollary 2.3 below). So, the set {% > 0}, which we shall identify as the set of allowed starting points, is open. Then our main result can be formulated as follows:

Theorem 1.1. Suppose that (H1) and (H2) with p := d + ε hold. Then there exists a diffusion process = (Ω, F, (Ft )t≥0 , (Xt )t≥0 , ( x)x∈{%>0} ) (i.e. a strong Markov process with continuous paths) with state space {% > 0} and cemetery ∆ := Alexandrov point of d whose transition semigroup (Pt )t>0 is Lr (µ)-strong Feller (i.e. Pt Lr (µ) ⊂ C({% > 0}), see Section 3), r ∈ [p, ∞), and which solves (1.1) in the (weak) sense for all initial conditions x ∈ {% > 0}. If (E, D(E)) is, in addition, conservative, then so is . Furthermore, (Pt )t>0 is strong Feller in this case (i.e. Pt (Bb ( d )) ⊂ Cb ({% > 0}) for all t > 0).

The process in Theorem 1.1 is constructed below directly and is not derived from the process ˜ associated to the Dirichlet form (E, D(E)). But the mere existence of ˜ solving (1.1) from µ − a.e. x ∈ d is used in a crucial way. Other main ingredients of the proof are results on elliptic regularity obtained by V.Bogachev, N.Krylov and the last named author of this paper in [BKR97] and [BKR01], as well as techniques from [Do02]. The organization of this paper is as follows: In Section 2 we recall the mentioned elliptic regularity results and present the consequences which are relevant for this paper. In Section 3 we construct the Lr (µ)-strong Feller semigroup (of kernels) and strong Feller resolvent (of kernels) for our diffusion process on {% > 0} which itself is constructed in Section 4. Of course, this process is then associated to (E, D(E)) given as the closure of the form (1.2). In Section 5 we prove that this process solves (1.1) in the sense of (Markov selections for) martingale problems (see [SV79]), hence in the weak sense. The first part of Section 6 is devoted to the said application to finite particle systems with singular interactions. Its second part treats diffusions in random media. Finally we should mention that this paper was strongly motivated by A.V.Skorohod’s paper [Sk99], in which he constructs solutions to (1.1) starting from x ∈ {% > 0} in the case of finite particle systems as we do in Subsection 6.1. His method is, however, based on Girsanov’s transformation and therefore requires assumptions on ∇% % which are much stronger than our assumptions (H1) and (H2) above.

2. An elliptic regularity result and its consequences Suppose that conditions (H1), (H2) (formulated in the Introduction) hold. Let (L2 , D(L2 )) be the generators of (E, D(E)) (cf. (1.2)) on L2 ( d , µ) (cf. [FOT94]), which is negative definite and self-adjoint and defined as the Friedrichs extension of its restriction L to C0∞ ( d ), (due to integration by parts) given by


4

(2.1)

Lu := ∆u + h

∇% , ∇ui d , u ∈ C0∞ ( %

d

).

Let Tt := etL2 , t > 0, and Gλ := (λ − L2 )−1 , λ > 0, be the corresponding C0 semigroup, resolvent on L2 ( d , µ), respectively. Then each Tt is sub Markovian, i.e. 0 ≤ f ≤ 1 ⇒ 0 ≤ Tt f ≤ 1, and so is each λGλ (see [FOT94]). It follows that both all Tt and all λGλ restricted to L1 ( d , µ) ∩ L∞ ( d , µ) extend to bounded operators on all Lr ( d , µ), r ∈ [1, ∞), with operator norms less than 1. We denote these extensions by the same symbols. (Tt )t>0 is then (because of its symmetry on L2 ( d , µ)) an analytic semigroup on every Lr ( d , µ), r ∈ (1, ∞). We refer e.g. to [LS96] for details on all this, in particular, Theorem 2.1 therein for the latter. Let (Lr , D(Lr )) denote the corresponding generators on Lr ( d , µ). Then for r ∈ [1, ∞) it is well-known and easy to see that C0∞ ( d ) ⊂ D(Lr ) and for all u ∈ C0∞ ( d )

(2.2)

Lr u = ∆u + h

∇% , ∇ui d , %

r d provided |∇%| , µ). % ∈ Ll◦c ( We need to prove regularity properties of (Tt )t>0 and (Gλ )λ>0 . We shall derive these from special cases of the elliptic regularity results in [BKR97] and [BKR01]. Let us restate these special cases, relevant for this paper.

Proposition 2.1. ([BKR97, Theorem 1(iii)(b)], [BKR01, Remark 2.15]) Let Ω be an open set in d and B = (B i ) : Ω → d , c : Ω → Borel measurable maps. Suppose µ is a (signed) Radon measure on Ω and f ∈ L1l◦c (Ω, dx) such that |B|, c ∈ L1l◦c (Ω, µ) and

Z

where

N u(x) µ(dx) =

Z

u(x)f (x) dx ∀ u ∈ C0∞ (Ω),

N u(x) := ∆u(x) + hB(x), ∇u(x)i + c(x)u(x). pd/(p+d)

If for some p > d, |B| ∈ Lpl◦c (Ω, µ), c ∈ Ll◦c then µ = % dx with % continuous and

1−d/p

1,p % ∈ Wl◦c (Ω, dx) (⊂ Cl◦c

1−d/p

pd/(p+d)

(Ω, µ), and f ∈ Ll◦c

(Ω, dx),

(Ω)),

where Cl◦c (Ω) denotes the set of all locally H¨ older continuous functions of order 1 − d/p on Ω. If Ω0 := Ω ∩ {% > 0} and moreover f, c ∈ Lpl◦c (Ω0 ), then for any open ball B ⊂ B ⊂ Ω0 there exists cB ∈ (0, ∞) (independent of % and f ) such that (2.3)

k%kW 1,p (B,dx) ≤ cB (k%kL1 (B,dx) + kf kLp(B,dx) ).

Corollary 2.2. Let % : d → 1,d+ε (H1), (H2). Then % ∈ Wl◦c ( 1−d/(d+ε) d dx-version in Cl◦c ( ).

d

be as specified in the Introduction, so satisfying , x) (with ε as in (H2)) and % has a continuous

Proof. For all u ∈ C0∞ (

d

) and µ := % dx integrating by parts we obtain from (2.1). Z Lu dµ = 0.

STRONG FELLER PROPERTIES |∇%| %

Since

d+ε ∈ Ll◦c (

d

5

, µ), the assertion follows by Proposition 2.1.

Below we shall always consider the continuous version of % and denote it also by %. Corollary 2.3. Assume (H1), (H2), set p := (d + ε) ∨ 2 (with ε as in (H2)), and let λ > 0. Suppose f ∈ Lr ( d , µ), r ∈ [p, ∞). Then

1,p % Gλ f ∈ Wl◦c (

d

, dx)

and for any open ball B ⊂ B ⊂ {% > 0} there exists cB,λ ∈ (0, ∞), independent of f , such that (2.4)

k% Gλ f kW 1,p (B,dx) ≤ cB,λ (kGλ f kL1 (B,µ) + kf kLp(B,µ) ).

Proof. Let us first assume that f ∈ C0∞ ( d ). Then Gλ f ∈ L∞ ( d , µ) ∩ L2 ( So, by the symmetry of L on L2 ( d , µ) we have that Z Z (λ − L)u Gλ f % dx = uf % dx ∀ u ∈ C0∞ ( d ).

d

, µ).

Now we may apply Proposition 2.1 with the measure ν := % Gλ f taking the role of µ to prove the assertion for f ∈ C0∞ ( d ). Since C0∞ ( d ) is dense in Lr ( d , µ) with respect to k kLr ( d,µ) , r ∈ [1, ∞), the assertion follows by continuity.

Corollary 2.4. Assume (H1), (H2), set p := (d + ε) ∨ 2 (with ε, as in (H2)) and let t > 0, r ∈ [p, ∞). (i) Let u ∈ D(Lr ). Then

1,p % Tt u ∈ Wl◦c (

d

, dx)

and for any open ball B ⊂ B ⊂ {% > 0} there exists cB ∈ (0, ∞) (independent of u and t) such that k% Tt ukW 1,p (B,dx)

(2.5)

≤ cB kTt ukL1 (B,µ) + kTt (1 − Lr )ukLp(B,µ) r−p r−1 ≤ cB µ(B) r kukLr ( d,µ) + µ(B) rp k(1 − Lr )ukLr (

(ii) Let f ∈ Lr (

d

d,µ)

.

, µ). Then the above statements still hold with (2.5) replaced by k% Tt f kW 1,p (B,dx)

(2.6)

≤ c˜B t−1 kf kLr (

d,µ)

,

where c˜B ∈ (0, ∞) (independent of f and t). Proof.

(i) We have that

r

Tt u = G1 Tt (1 − Lr )u.

Since Tt (1 − Lr )u ∈ L ( d , µ), the assertion follows by Corrollary 2.3. (ii) (Tt )t>0 is an analytic semigroup on Lr ( d , µ), and since (2.6) holds by (i) for all f ∈ C0∞ ( d ) which are dense in Lr ( d , µ), the assertion follows by continuity.


6

Remark 2.5. We note that since by Sobolev W 1,p (B, dx) ⊂ C 1−d/p (B, dx) continuously, it follows by (2.5) for r ∈ [p, ∞), R > 0, that the set {Tt u | t > 0, u ∈ D(Lr ), kukLr (

d,µ)

+ kLr ukLr (

d,µ)

≤ R}

is equicontinuous on {% > 0}. 3. Construction of the semigroup and resolvent of kernels Assume throughout this section that (H1) and (H2) hold and let p := (d + ε) ∨ 2 with ε as in (H2). Below for a topological space X with Borel σ-algebra B(X) we denote the set of all B(X)-measurable f : X → which are bounded, or nonnegative by Bb (X), B + (X) respectively. r For the set of all B( d )-measurable functions such R r ∈r [1, ∞) let L (µ) denote r that |f | dµ < ∞. For f ∈ L (µ) and an operator T on Lr ( d , µ) we write T f in the sense that T is applied to the corresponding µ-class in Lr ( d , µ) provided this class is in the domain of T . For t > 0 and f ∈ Lp (µ) we know by Corollary 2.4(ii) and Sobolev embedding (cf. Remark 2.5) that Tt f has a (real-valued and) continuous, hence unique µ-version g T t f on {% > 0}. Furthermore, we have:

Lemma 3.1. Let t > 0 and x ∈ {% > 0}. Then the map g f 7→ T t f (x)

on Lp (µ) is a Daniell-Integral, hence there exists a unique positive measure P t (x, dy) on B( d ) such that Z g T f (y) Pt (x, dy) ∀ f ∈ Lp (µ). t f (x) =

g Proof. Since the map f 7→ T t f is positive and linear µ−a.e. on {% > 0}, it is positive and linear pointwise on {% > 0} by continuity. Furthermore, if fn ∈ Lp (µ), n ∈ , such that fn ↓ 0, then by (2.6) (applied with r := p) and Sobolev embedding (cf. Remark 2.5) Tg t fn (x) ↓ 0. So, we can apply the Daniell-Stone theorem.

As usual we define for t > 0, x ∈ {% > 0}, and f ∈ L1 ( Z (3.1) Pt f (x) := f (y) Pt (x, dy)

d

; Pt (x, dy)) ∪ B + (

d

)

and P0 f (x) := f (x).

Proposition 3.2. (i) Let t > 0. Then Pt 1(x) ≤ 1 for all x ∈ {% > 0} and there exists a B({% > 0} × d )-measurable map (x, y) 7→ pt (x, y) such that Pt (x, dy) = pt (x, y) µ(dy). In particular, Pt (x, {% = 0}) = 0 for all x ∈ {% > 0}, so Pt can be considered as a kernel on {% > 0} (denoted below by the same symbol). (ii) (Pt )t>0 is a semigroup of kernels on {% > 0} which is Lr (µ)-strong Feller for S all r ∈r [p, ∞), i.e. (cf. [Do02]) Pt f ∈ C({% > 0}) for all t > 0, f ∈ r∈[p,∞) L (µ). (iii) For all u ∈ C0∞ ( d ) and all s ≥ 0

lim Pt+s u(x) = Ps u(x) ∀ x ∈ {% > 0}.

t→0


(iv) (Pt )t>0 is a measurable semigroup on {% > 0}, i.e. for f ∈ B + ( (t, x) 7→ Pt f (x) is B([0, ∞) × {% > 0})-measurable.

7 d

) the map

Proof. (i) Let Kn , n ∈ , be compact sets in d such that Kn ↑ d . Then because Tt 1Kn ≤ 1 µ − a.e. for all n ∈ , we have for all x ∈ {% > 0} by continuity that Pt 1Kn (x) ≤ 1 for all n ∈ . Hence

Pt 1(x) = lim Pt 1Kn (x) ≤ 1 ∀ x ∈ {% > 0}. n→∞

Let N ∈ B( d ), µ(N ) = 0. Then Pt 1N = 0 µ − a.e. on {% > 0}, so by continuity everywhere on {% > 0}. So, the existence of pt (x, y) follows. That it can be chosen measurable in both arguments is standard. (ii) Let s, t > 0. Then for all u ∈ C0∞ ( d ), since Tt+s u = Tt Ts u µ − a.e.,

Pt+s u(x) = Pt (Ps u)(x) for µ − a.e. x ∈ {% > 0}. Since u, Ps u ∈ Lp (µ), both sides of the equality are continuous, so it holds for all x ∈ {% > 0}. Now the first assertion follows by a monotone class argument. To show the second let t > 0, r ∈ [p, ∞) and f ∈ Lr (µ). Since f = f + − f − , we may assume that f ≥ 0. Let χn ∈ C0∞ ( n ), χn ≥ 0, for n ∈ such that χn ↑ 1 as n → ∞. Set fn := χn f . Then Pt fn ↑ Pt f pointwise on {% > 0} as n → ∞. But since fn → f in Lr ( d , µ) as n → ∞, by (2.6) and Sobolev embedding we also have that (Pt fn )n∈ has a continuous pointwise limit on {% > 0}. So, Pt f must be this limit. (iii) Let u ∈ C0∞ ( d ). Then by Remark 2.5 and the definition of Pt , t > 0, the set {Pt (Ps u) | t > 0} is equicontinuous on {% > 0}. (Note that Ps u ∈ D(Lp ) since u ∈ D(Lp )). Suppose x ∈ {% > 0} such that for some δ > 0 there exists a sequence (tn )n∈ tending to zero such that

(3.2)

|Ptn (Ps u)(x) − Ps u(x)| ≥ δ ∀ n ∈ . Since Ttn Ts u → Ts u in Lp ( d , µ) as n → ∞, (selecting another subsequence, if necessary) we may assume that for n o M := y ∈ {% > 0} : lim Ptn (Ps u)(y) = Ps u(y)

n→∞

we have µ({% > 0} \ M ) = 0. So, M is dense in {% > 0}. Since {Ptn (Ps u) | n ∈ } is equicontinuous, it follows that lim Ptn (Ps u)(y) = Ps u(y) ∀ y ∈ {% > 0}

n→∞

contradicting (3.2). (iv) This is a consequence of (iii) by a monotone class argument. Remark 3.3. We would like to point out that it is not S claimed above that Pt f for t > 0, is continuous for f ∈ Bb ( d ), but only for f ∈ r∈[p,∞) Lr (µ). So, (Pt )t>0 is not strong Feller in the classical sense. For this reason we shall have a closer look at the corresponding resolvent for which this is the case. We emphasize that all “strong Feller statements” below by Corollaries 2.3 and 2.4 always state that 1−d/p 1,p Pt f, Rλ f are not only continuous, but even in Wloc ( d , dx) ⊂ Cloc ( d ) for the respectively specified functions f .


8

For λ > 0, f ∈ Lp (µ) we know by Corollary 2.3 that Gλ f has a unique (real-valued g and) continuous version G λ f on {% > 0}. Hence because of (2.4) exactly as in Lemma 3.1 one proves: Lemma 3.4. Let λ > 0 and x ∈ {% > 0}. Then the map g f 7→ G λ f (x)

on Lp (µ) is a Daniell-integral, hence there exists a unique positive measure R λ (x, dy) on B( d ) such that Z g Gλ f(x) = f (y) Rλ (x, dy) ∀ f ∈ Lp (µ)

As usual we define for λ > 0, x ∈ {% > 0}, and f ∈ L1 ( Z (3.3) Rλ f (x) := f (y) Rλ (x, dy).

d

, Rλ (x, dy)) ∪ B + (

d

)

Proposition 3.5. (i) Let λ > 0 and x ∈ {% > 0}. Then λRλ 1(x) ≤ 1 and there exists a B({% > 0} × d )-measurable map (x, y) 7→ rλ (x, y) such that Rλ (x, dy) = rλ (x, y) µ(dy). In particular, Rλ (x, {% = 0}) = 0, so Rλ can be considered as a kernel on {% > 0} (denoted below by the same symbol). (ii) (Rλ )λ>0 is a resolvent of kernels on {% > 0}. (iii) (Rλ )λ>0 is Lr (µ)-strong Feller for all r ∈ [p, ∞], i.e. Rλ f ∈ Cb ({% > 0}) for all f ∈ Bb ( d ), and Rλ f ∈ C({% > 0}) for all λ > 0 and all f ∈ S r r∈[p,∞) L (µ). (iv) Let λ > 0. Then for all f ∈ Bb ( d ) ∪ B + ( d ) and all x ∈ {% > 0} Z ∞ (3.4) e−λt Pt f (x) dt. Rλ f (x) =

0

(v) For all u ∈

C0∞ ( d )

lim λRλ u(x) = u(x) ∀ x ∈ {% > 0}.

λ→∞

Proof. The proofs of (i), (ii) are because of (2.4) analogous to the corresponding statements in Proposition 3.2. S (iii) Let λ > 0 and f ∈ Bb ( d )∪ r∈[p,∞) Lr (µ). Since f = f + −f − , we may assume that f ≥ 0. Then for fn , n ∈ , defined as in the proof of Proposition 3.2(ii), we have Rλ fn ↑ Rλ f pointwise on {% > 0} as n → ∞. But since Rλ fn → Rλ f in L1 (B, µ) and fn → f in Lp (B, µ) as n → ∞ for every ball B ⊂ B ⊂ {% > 0}, by (2.4) and Sobolev embedding we also have that (Rλ fn )n∈ has a continuous pointwise limit on {% > 0}. So, Rλ f must be this limit, which is bounded by (i), if f ∈ Bb ( d ). (iv) It suffices to consider f ∈ C0∞ ( d ). Since Gλ f is the Laplace transform of Tt f, t > 0, in L2 ( d , µ), it follows that (3.4) holds for µ − a.e. x ∈ {% > 0}. Since both sides of equality (3.4) are continuous on {% > 0}, the assertion follows. (v) Transforming the integral in (3.4), the assertion follows from (iv) by Proposition 3.2(iii), and Lebesgue’s dominated convergence theorem.


9

Lemma 3.6. (i) Let t, λ > 0 and f ∈ B + ( d ). Then Pt f (resp. Rλ f ) is lowersemicontinuous on {% > 0}. If g ∈ B + ( d ) such that f ≤ g and Pt g (resp. Rλ g) is (real-valued and) continuous on {% > 0}, then so is Pt f (resp. Rλ f ). (ii) Let λ > 0, t > 0. Then Pt f is continuous on {% > 0} for all B( d )-measurable f : d → such that f ≤ cR1 1 for some c ∈ (0, ∞).

Proof. (i) We shall prove the assertion for Pt . The proof for Rλ is exactly the same. There exist fn ∈ Lp (µ), fn ≥ 0, n ∈ , such that fn ↑ f . Hence Pt f is lower semicontinuous on {% > 0}, since each Pt fn is continuous there. Hence also Pt (g − f ) is lower-semicontinuous, so Pt f = Pt g − Pt (g − f )

is continuous on {% > 0}. (ii) Since Pt 1 ≤ 1, it follows by Proposition 3.5(iii) that Rλ Pt 1 is continuous on {% > 0}. But clearly by (3.4), Rλ Pt 1 = Pt Rλ 1 on {% > 0}, hence the assertion follows by (i). Let us now assume conservativity of (E, D(E)) on L2 ( d , µ), more precisely: (H3) For the adjoint semigroup (Tt∗ )t>0 of (Tt )t>0 considered as a C0 -semigroup on L1 (E, µ), i.e. (Tt∗ )t>0 is a (non-C0 ) semigroup on L∞ ( d , µ), we have Tt∗ 1 = 1 (µ − a.e.) for all t > 0.

Remark 3.7. (i) There is a complete (analytic) characterization of (H3) in terms of properties of % by W.Stannat in [St99] to which we refer for details, in particular [St99, Corollary 2.2 and Proposition 1.10]. We would only like to mention here that Stannat, in particular, proved that (H3) is equivalent to the so-called L1 -uniqueness, i.e. the property that C0∞ ( d ) is a core for the generator (L1 , D(L1 )) of (Tt )t>0 on L1 ( d , µ) (cf. [St99, Corollary 2.2 and Remark 2.4]). E.g. if µ( d ) < ∞, then (H3) holds. (ii) For global Lp -estimates on the resolvent under global assumptions (on the coefficients) we refer to [Fu77].

Proposition 3.8. If (in addition to (H1), (H2) also) (H3) holds, then: (i) λRλ 1(x) = 1 for all x ∈ {% > 0}, λ > 0. (ii) (Pt )t>0 is strong Feller on {% > 0}, i.e. Pt (Bb ( d )) ⊂ Cb ({% > 0}) for all t > 0. (iii) Pt 1(x) = 1 for all x ∈ {% > 0}, t > 0.

Proof. (i) Let Kn ⊂ d , n ∈ , be compact. Then for t > 0 and g ∈ L1 ( L∞ ( d , µ), by the symmetry of Tt on L2 ( d , µ) Z Z Z g dµ = Tt∗ 1 g dµ = Tt g dµ Z (3.5) = lim 1Kn Tt g dµ n→∞ Z Z = lim Pt 1Kn g dµ = Pt 1 g dµ.

d

, µ) ∩

n→∞

Multiplying both sides of (3.5) by e−λt for λ > 0, and integrating with respect to dt over [0, ∞), by Fubini’s theorem we obtain that λRλ 1 = 1 µ−a.e., hence by continuity we obtain (i).


10

(ii) Let f ∈ Bb ( d ) and c := supx∈ d f (x). Then by (i) we have that f ≤ cR1 1 and the assertion follows by Lemma 3.6(ii). (iii) Let t > 0. It follows by (3.5) that Pt 1 = 1 µ − a.e. But by (ii), Pt 1 is continuous on {% > 0}. So, (iii) follows.

4. Construction of the associated diffusion process Throughout this section we assume (H1) and (H2) and (as before) denote the Alexandrov compactification of d by d∆ . We extend our semigroup of kernels (Pt )t>0 defined in Section 3 as kernels from {% > 0} to B( d ) to a semigroup on B( d∆ ) in the following standard way: for x ∈ {% > 0}, t > 0, extend Pt (x, dy) to a probability measure Pt∆ (x, dy) on B( d∆ ) (= {A ⊂ d∆ | A ∈ B( d ) or A = A0 ∪ {∆}, A0 ∈ B( d )}) by setting

Pt∆ (x, dy) := (1 − Pt (x,

(4.1)

d

))ε∆ (dy) + Pt (x, dy)

and for x ∈ {% = 0} ∪ {∆} define

Pt∆ (x, dy) := εx (dy),

(4.2)

where εx denotes Dirac measure at x. It is straightforward to check that (Pt∆ )t>0 is a semigroup of probability kernels on B( d∆ ). We extend µ to B( d∆ ) by zero. For n ∈ , set [ Sn := {k2−n | k ∈ ∪ {0}} and S := Sn .

n∈

By Kolmogorov’s standard construction scheme there exist probability measures d d S 0 x, x ∈ ∆ , on Ω := ( ∆ ) , equipped with the product σ-field F , so that 0 0 0 0 := (Ω, F , (Fs )s∈S , (Xs )s∈S , ( x )x∈ d∆ ) is a normal Markov process on d∆ with transition semigroup (Ps∆ )s∈S . Here Xs0 : ( d∆ )S → d∆ are the coordinate maps, and Fs0 := σ(Xr0 | r ∈ S, r ≤ s). Define Z (4.3) µ := x µ(dx).

d ∆

There exists a complete metric on d∆ compatible with the Alexandrov topology of d∆ . Our aim is to find a set Ω0 ∈ F 0 such that x (Ω0 ) = 1 for all x ∈ {% > 0} and each ω = (ω(s))s∈S has a unique extension to a continuous path in {% > 0} ∪ {∆} (equipped with the trace topology induced by the Alexandrov topology on d∆ ) which stays in ∆ once it gets there. Now (with the aim to construct Ω0 as specified above) we are going to use the theory of symmetric Dirichlet forms, in particular, the Hunt process ˜ = ˜ F, ˜ (F˜t )t≥0 , (X ˜ t )t≥0 , ( ˜ x )x∈ d), with Ω ˜ as in (1.3), associated to (E, D(E)), which (Ω, was introduced in the Introduction. We shall follow the notation and terminology of [FOT94] without repeating everything here. The following result is crucial for our further analysis. It holds even merely under (H1).

Lemma 4.1. {% = 0} is of capacity zero with respect to (E, D(E)). Proof. See [Fu84], if (H1), (H2) holds. If merely (H1) holds, see [Ho95, Theorem 3].


11

By virtue of e.g. [MR92, Chap. IV, Theorem 5.4 and Chap. III, Exercise 2.10] Lemma 4.1 implies that there exists an increasing sequence (Kn )n∈ of compact subsets of {% > 0}, so that for the first hitting times σKnc of their complements ˜ t ∈ K c }, n ∈ , and Knc := d \ Kn , i.e., σKnc := inf{t > 0 | X n n o ˜ 0 := ω ∈ Ω ˜ | ω(0) ∈ {% > 0}, lim σK c ≥ ζ , (4.4) Ω n

n→∞

we have

(4.5)

˜ µ (Ω ˜ 0 ) = 1,

where (4.6)

˜ µ :=

Z

˜ x µ(dx).

d

˜ → Ω defined by Consider the one-to-one map G : Ω

˜ G(ω) := (ω(s))s∈S , ω = (ω(t))t∈[0,∞) ∈ Ω.

(4.7)

˜ s | s ∈ S}, then obviously Ω ˜ 0 ∈ F˜0 and G is F˜ 0 /F 0 Note that if F˜ 0 := σ{X measurable. Define the image measure of the restriction of ˜ µ to F˜ 0 of ˜ µ (defined in (4.7)) under G by ˆ µ := ˜ µ ˜ 0 ◦ G−1 . (4.8) F

Lemma 4.2.

ˆµ =

µ.

Proof. Let N ∈ , A0 , . . . , AN ∈ B( d∆ ), s1 , . . . , sN ∈ S, 0 ≤ s1 ≤ . . . ≤ sN . Let (P˜t )t>0 be the transition semigroup of ˜ and define P˜t∆ (x, dy) for x ∈ d as in (4.1) and P˜t∆ (∆, dy) := ε∆ . Then since ˜ is associated to (E, D(E)), i.e. P˜t f is a µ-version Tt f for all f ∈ L2 (µ), t > 0, and since µ({% > 0}) = 0 we obtain by definition of ˆ µ that

ˆ µ [X 0 ∈ A0 , X 0 ∈ A1 , . . . , X 0 0 s1 s Z N = 1 A0 Z = 1 A0 =

hence ˆ µ =

µ

0 µ [X0

∈ AN ] P˜s∆1 1A1 P˜s∆2 −s1 1A2 · · · 1AN −1 P˜s∆N 1AN dµ Ps∆1 1A1 Ps∆2 −s1 1A2 · · · 1AN −1 Ps∆N 1AN ∈ A0 , Xs01 ∈ A1 , . . . , Xs0N ∈ AN ],

dµ

on F 0 .

Lemma 4.3. ˜ 0 ) ∈ F 0 and G(Ω

˜

µ (G(Ω0 ))

= 1.

Proof. Since G is one-to-one and F˜ 0 /F 0 -measurable, the first assertion follows, since F 0 is countably generated and F˜ 0 is standard Borel (cf. [Pa67, Theorem 2.4, p. 135]). The second statement follows, since by Lemma 4.2, (4.8) and (4.5)

˜

µ (G(Ω0 ))

˜ 0 )) = ˜ µ (Ω ˜ 0 ) = 1. = ˆ µ (G(Ω


12

˜ 0 ) the desired set Ω0 of x-measure for one for all x ∈ {% > To construct from G(Ω 0}, we follow the strategy of [Do02] and define \ ˜ 0 )), Ω1 := (4.9) θs−1 (G(Ω

s∈S, s>0

where θs : Ω → Ω, θs (ω) := ω(· + s), for s ∈ S, is the usual time shift operator. Lemma 4.4. Suppose A ∈ F 0 , such that (4.10)

−1 x (θs (A))

In particular,

x (Ω1 )

µ (A)

= 1. Then for every s ∈ S, s > 0,

= 1 ∀ x ∈ {% > 0}.

= 1 for all x ∈ {% > 0}.

Proof. Let x ∈ {% > 0}. Then by the Markov property and Proposition 3.2(i) (4.11)

x (Ω

\ θs−1 (A)) =

−1 x (θs (Ω

\ A))

◦ θs | Fs0 ) = x Xs0 (1Ω\A )

=

x

x (1Ω\A

= Ps ( · (Ω \ A))(x) Z = y (Ω \ A) ps (x, y) µ(dy)

= 0.

Here x ( · ), x ( · | Fs0 ) denotes expectation, conditional expectation with respect to x respectively. This proves the first assertion, the second then follows by Lemma 4.3.

Remark 4.5. Lemma 4.4 will be a key ingredient of our line of arguments below. In its proof we used the absolute continuity of Pt (x, dy) with respect to µ for x ∈ {% > 0}, rather than the strong Feller property. This is in contrast to the corresponding results in [DPR01] and [Do02]. But it is in accordance with general results in the theory of symmetric Dirichlet forms that a number of statements valid for all x ouside a capacity zero set, turn into statements for all x in the state space, if Pt (x, dy) has a density with respect to the underlying measure (cf. [FOT94]). Obviously, Ω1 defined in (4.9) consists of paths in Ω which have unique continuous extensions to (0, ∞) which still lie in {% > 0} ∪ {∆} and stay in ∆ once they have hit ∆. So, we have to handle the limits at s = 0. To this end we define n o Ω0 := ω ∈ Ω1 lim Xs0 (ω) exists in {% > 0} . (4.12) s↓0

We shall see that Ω0 is our desired set.

Lemma 4.6. Let x ∈ {% > 0}. Then

lim Xs0 = x

(4.13)

s↓0

Proof. Let un ∈ C0∞ ( define

d

), n ∈

x

− a.s.

, separating the points of

d ∆.

fn,1 := ((1 − L)un )+ , fn,2 := ((1 − L)un )− .

Fix n ∈

and


13

Since (1 − L)un ∈ Lp (µ), it follows by Proposition 3.5(iii) that R1 (fn,1 ), R1 (fn,2 ) are both (real-valued and) continuous on {% > 0}. Furthermore (as is well known and easily follows from the Markov property), (e−s R1 fn,i (Xs0 ))s∈S , i = 1, 2 , are positive supermartingales, so by the martingale convergence theorem x − a.s.

lim e−s R1 fn,i (Xs0 ) exists in

+

s↓0

so

x

− a.s. on Ω1

lim un (Xs0 ) exists in

(4.14)

s↓0

for i = 1, 2,

∀n∈ ,

since un = R1 fn,1 − R1 fn,2 on {% > 0} for all n ∈ . But by Proposition 3.2(iii) for all n ∈ 0 2 = Ps u2n (x) − 2un (x)Ps un (x) + u2n (x) → 0 x (un (Xs ) − un (x))

as s ↓ 0, which together with (4.14) implies that

x

− a.s. on Ω1

lim un (Xs0 ) = un (x) ∀ n ∈ . s↓0

Since un , n ∈ , separate the points of by Lemma 4.4.

d ∆,

this implies (4.13), since

x (Ω1 )

=1

Now we define for ω ∈ Ω and t ≥ 0 ( lim Xs0 (ω) , if ω ∈ Ω0 s↓t Xt (ω) := (4.15) s∈S x0 , if ω ∈ Ω \ Ω0 , where x0 is a fixed point in {% > 0}. Now we can formulate the final result of this section. Theorem 4.7. There exists a diffusion process (i.e. strong Markov with con= (Ω, F, (Ft )t≥0 , (Xt )t≥0 , ( x )x∈{%>0} ) with state space tinuous sample paths) {% > 0} and cemetery ∆ := the Alexandrov point of d , having as transition semigroup the strong Lr (µ)-Feller semigroup (Pt )t>0 , r ∈ [p, ∞) (defined in Lemma 3.1 and (3.1)).

Proof. Defining Xt , t ≥ 0, as in (4.15) and (Ft )t∈[0,∞] as the corresponding natural filtration (see e.g. [MR92, Chap. IV, Definition 1.8, formulae (1.6), (1.7)]), the proof is standard. We only note that for u ∈ C0∞ ( d ) and t > 0, x ∈ E, Z Z u(X 0 ) − u(Xt ) 2 d x = lim u(X 0 ) − u(X 0 ) 2 d x t s t

s∈S s↓t

= lim (pt u2 (x) − 2pt (ups−t u)(x) + ps u2 (x)) s∈S s↓t

=0

by Proposition 3.2(iii). Hence

0 x [Xt

6= Xt ] = 0.

Remark 4.8. If, in addition, (H3) holds, one can drop ∆ in the above theorem, and one obtains a conservative diffusion process.


14

5. Solution to the stochastic equation Throughout this section we assume (H1), (H2) to hold. Our aim is to prove from Theorem 4.7 gives rise to a weak (= martingale) that the diffusion process solution to the stochastic equation (1.1) and is unique, in a sense specified below. So, let x, x ∈ {% > 0}, be as in Theorem 4.7. As usual below we extend every function f : {% > 0} → to {% > 0}∪{∆} by setting f (∆) = 0. Thus, we can avoid to explicity mention the explosion time (= life time ζ from the previous section), i.e. the time the process leaves any compact set. The following lemma is crucial to prove that each x solves the martingale problem for (L, C0∞ ({% > 0})) with the initial condition x ∈ {% > 0}.

Lemma 5.1.

(i) Let f ∈

S

r∈[p,∞)

Z

Lr (µ), f ≥ 0, then for all t > 0, x ∈ {% > 0},

t 0

Pr f (x) dr < ∞,

hence ZZ (ii) Let u ∈ C0∞ ( (5.1)

Proof.

f (Xr ) dr d 0

x

0. Then

Rλ ((λ − L)u)(x) = u(x) for all x ∈ {% > 0}.

(iii) Let u ∈ C0∞ ( (5.2)

d

t

d

), t > 0. Then

Pt u(x) − u(x) =

Z

t 0

Pr (Lu)(x) dr for all x ∈ {% > 0}.

(i) Let t > 0, x ∈ {% > 0}. By (3.4) and monotone convergence R1 f (x) =

Z

∞

e−r Pr f (x) dr.

0

S But R1 f is continuous, hence finite on {% > 0}, since f ∈ r∈[p,∞) Lr (µ). So, assertion (i) follows. (ii) Since Gλ (λ − L)u = u in Lp ( d , µ), it follows that (5.1) holds for µ-a.e. x ∈ {% > 0}. Since (λ − L)u ∈ Lp (µ), we have that Rλ ((λ − L)u) is continuous on {% > 0} and so assertion (ii) follows. (iii) Note that since Lu ∈ Lp (µ) the integral on the right hand side of (5.2) exists Rt and is a µ-version of 0 Tr (Lu) dr, as is the left hand side of Tt u − u. Hence (5.2) holds for µ-a.e. x ∈ {% > 0}. Since the left hand side of (5.2) is continuous on {% > 0}, the assertion follows if we can prove that so is the right hand side. But by (ii) and (3.4), which extends to all f ∈ Lp (µ), we


15

have for all x ∈ {% > 0} Z t Pr (Lu)(x) dr 0 Z r r=t Z t Z r e−s Ps (Lu)(x) ds = er e−s Ps (Lu)(x) ds dr − er r=0 0 0 0 Z ∞ = et R1 (Lu)(x) − e−s Ps (Lu)(x) ds t Z ∞ Z t e−s Ps (Lu)(x) ds dr er R1 (Lu)(x) − − r

0

= et [R1 (Lu)(x) − e−t Pt (R1 (Lu))(x)] Z t t − (e − 1)R1 (Lu)(x) − Pr (R1 (Lu))(x) dr. 0

We see that, since all involved functions are in Lp (µ), all functions in the last expression are obviously continuous in x ∈ {% > 0} apart from the last. But by Fubini’s theorem for f := (Lu)+ or (Lu)− Z t Z t Pr (R1 f )(x)dr = R1 ( Pr f (·)dr)(x). 0

0

Rt But 0 Pr f (·) dr ∈ L (µ) since it is a µ-version of 0 Tr f dr which is in Lp ( d , µ). So, also this last term is continuous on {% > 0}. So, assertion (iii) is proved.

Rt

p

Theorem 5.2. For every x ∈ {% > 0}, x from Theorem 4.7 solves the martingale problem for (L, C0∞ ({% > 0})) with initial condition x, i.e. under x for all u ∈ C0∞ ({% > 0})

(5.3)

u(Xt ) − u(x) −

Z

t 0

Lu(Xs ) ds, t ≥ 0,

is an (Ft )t≥0 -martingale starting at zero. Proof. By Lemma 5.1 the proof is now standard by the Markov property. All integrability issues in the computations are clear from Lemma 5.1. Now we turn to uniqueness. Definition 5.3. A diffusion process 0 = (Ω0 , F 0 , (Ft0 )t≥0 , (Xt0 )t≥0 , ( 0x )x∈{%>0} ) on {% > 0} with lifetime ζ 0 , cemetery ∆, and semigroup (Pt0 )t>0 is said to satisfy the L1 ({% > 0}, µ)-martingale problem for (L, C0∞ ({% > 0})), if (i) For some M 0 , ε0 ∈ (0, ∞) Z Z |Pt0 f | dµ ≤ M 0 |f | dµ ∀f ∈ Cb ({% > 0}), t ∈ (0, ε0 ). E E R 0 ∞ (ii) For all u ∈ C0 ({% > 0}) under 0µ := x µ(dx) Z t u(Xt0 ) − Lu(Xs0 ) ds, t ≥ 0,

0

is an (Ft0 )t≥0 -martingale.


16

from Theorem 4.7 solves the L 1 ({% > Proposition 5.4. The diffusion process ∞ 0}, µ)-martingale problem for (L, C0 ({% > 0})).

Proof. The proof is completely analogous to that of Proposition 8.2 in [DPR01]. Theorem 5.5. (“Uniqueness”) Assume (in addition to (H1), (H2)) that (H3) holds. Let 0 = (Ω0 , F 0 , (Ft0 )t≥0 , (Xt0 )t≥0 , ( 0x )x∈{%>0} ) be a diffusion process on {% > 0} with transition semigroup (Pt0 )t>0 such that 0 satisfies the L1 ({% > 0}, ν)martingale problem for (L, C0∞ ({% > 0})). Then 0x = x for µ − a.e. x ∈ {% > 0}, in Theorem 4.7. If, in where x, x ∈ {% > 0}, is the probability measure of addition, Pt0 (C0∞ ({% > 0})) ⊂ C({% > 0}) for all t > 0, then 0x = x for every x ∈ {% > 0}.

Proof. As mentioned in Remark 3.7, by [St99] the conservativity (H3) implies L 1 uniqueness of (L, C0∞ ({% > 0})). Hence the proof of the assertion is completely analogous to that of Theorem 8.3 in [DPR01]. Remark 5.6. In a standard way one derives from Theorem 5.2 that (Xt )t≥0 under x weakly solves the stochastic equation (1.1) for all starting points x ∈ {% > 0}, up to the explosion time (= lifetime ζ), i.e. up to the time the process leaves any compact set. The only thing to do is to calculate the quadratic variation of the martingale in (5.3) to be equal to Z t |∇u|2 (Xs ) ds, t ≥ 0,

0

under x for all x ∈ {% > 0}. This can be done directly by a little lengthy calculation with serveral integrability issues to be solved on the basis of Lemma 5.1 and the arguments in its proof (cf. [RS02]). By P.Levy’s characterization theorem and a localization argument one then obtains (1.1). We would like to stress that one cannot use the Fukushima decomposition to get (1.1) for , as is usually done (cf. [Fu82a, Fu82b], [AR91] and also [FOT94]), because this would give (1.1) only for x ∈ E outside a capacity zero set. More refined results from [FOT94] to give (1.1) for every x do not seem to apply directly, since our transition semigroup is only defined for x ∈ {% > 0} and has only a density (x, y) 7→ pt (x, y) with respect to µ for such x. But we are convinced that also by these refined results, properly modified, we can derive (1.1) for our and all x ∈ {% > 0}.

6. Applications to stochastic dynamics In this chapter we describe two classes of models from Mathematical Physics in which the stochastic equations of the form (1.1) with singular drifts ∇% % appear naturally. In our considerations we do not try to analyze these models under most the general assumptions. We rather restrict ourselves to some particular situations which are reasonable from the physical point of view and give an illustration of the effectiveness of our general results in applications. More detailed and extended analysis of these models will be a subject of the forthcoming paper [AKR02]. 6.1. Stochastic gradient dynamics of N -particle systems. Let us consider a system of N particles in the Euclidean space d , d ≥ 2, which have positions xk ∈ d , 1 ≤ k ≤ N . The particles interact via a pair potential V : d {0} → .


17

The stochastic motion of the particles is described by the system of stochastic differential equations (SDE) dxk (t) = −

(6.1)

N X

j=1,j6=k

xk (0) = xk ∈

d

∇V (xk (t) − xj (t)) dt +

, 1 ≤ k ≤ N,

√

2dwk (t)

where {xk , 1 ≤ k ≤ N } are different points in d and {wk , 1 ≤ k ≤ N } are independent standard Wiener processes in d . Following [Sk96] (see also [Sk99]) we will call the system regular if in the stochastic motion particles never can hit each other. We assume that the potential V has the form V (x) = φ(|x|), x ∈ d , where φ ∈ C 2 ( + \ {0}) and this function and its first and second derivatives φ(1) , φ(2) have the following asymptotic properties: A1 (6.2) , α1 , A1 > 0, r → 0+, φ(2) (r) α r 1 +2

A2 , α2 > d, A2 > 0, s = 0, 1, 2, r ≥ r0 > 0. rα2 +s We should admit a singularity of φ at 0 as a natural condition from the point of view of Mathematical Physics. If α1 < d−1, i.e. the singularity of φ is not very big, then the system (6.1) has a unique strong solution and this solution is regular, see [Sk96, Sk99]. But the physically important case of a superstable potential (α1 > d) is hence not covered and needs the additional analysis provided by this paper. To include this case in the general framework of the paper we introduce the following potential energy functional X E (x) := V (xk − xj ) (≤ +∞) , x = (x1 , ..., xN ) ∈ dN ,

(6.3)

|φ(s) (r) | ≤

1≤k 0} = {x = (x1 , ..., xN ) ∈

and obviously % ∈

and

dN

Cb2 ( dN ).

| xk 6= xj if k 6= j},

Note that now 1 √ % (x) = exp − E (x) , % ∈ Cb2 2

p

N  X ∇% ∇V (xk − xj ) (x) = − % j6=k

∈

dN

dN

, x ∈ {% > 0}.

k=1

It is not hard to show that |∇%| (6.4) ∈ Lploc dN , µ , p ≥ 1, % where as above µ (dx) = % (x) dx. Therefore, both assumptions (H1), (H2) above are satisfied. Moreover, the boundedness of the density % implies that the conservativity assumptions (H3) is also true, see Remark 3.7(i) and e.g. [Li99], Theorem 4. Then


18

all results above can be applied to the system under consideration. In particular, for any order of singularity of V at 0 ∈ d there exists a conservative diffusion in SN which gives the unique solution to the martingale problem associated with the system (6.1). This system is hence regular in the sense of [Sk96, Sk99]. The question whether this process gives a strong solution to (6.1) remains open.

6.2. Diffusions in a random media. Consider now another model in which a particle performs a random motion in the Euclidean space d , d ≥ 2, interacting with randomly distributed impurities. This model can be formalized as follows. The impurities form a locally finite subset (i.e. configuration) γ = {xk |k ≥ 1} ⊂ d and the interaction between the moving particle and particles from γ is given by the pair potential V as in Subsection 6.1. To simplify our considerations we assume instead of (6.3) the stronger condition that φ has finite support (i.e. we have a finite range of the interaction). The configurations γ are distributed according to a given random point process on d . In mathematical physics this random point process usually corresponds to a Gibbs measure ν on the configuration space over d . The stochastic dynamics of the considered particle is described by the following SDE: ∞ X √ (6.5) dξ (t) = − ∇V (ξ (t) − xk ) dt + 2dw (t) ,

k=1

ξ (0) = x ∈

d

\ γ,

where w is the standard Wiener process in d . This equation describes a diffusion process with a random drift of a special type. For a review on the stochastic dynamics in random velocity fields see, e.g., [Ol94]. Essential difficulties in the study of the solution to (6.5) are related not only to the singularity of the potential V but also to an additional irregularity of the drift term in (6.5) coming from the configuration γ. To be able to control the drift in (6.5) we will restrict the class of admissible configurations. Let B (x, r) := {y ∈ d | |y − x| < r} denote the open ball of radius r > 0 with center at x ∈ d . Define the set U of all configurations γ in d such that

∀r > 0 ∃c (γ, r) > 0 : (6.6)

|γ ∩ B (x, r)| ≤ c (γ, r)

p log (1 + |x|) , x ∈

d

,

where |A| denotes the cardinality of a set A. Note that for many classes of probability measures ν on the configuration spaces we have ν (U ) = 1, see [KKK02]. In particular, this is true for the well-known Ruelle measures corresponding to pair superstable potentials [KY93]. We introduce the potential energy of the particle in the configuration γ ∈ U as a function Eγ : d → of the form

Eγ (x) :=

∞ X k=1

Then for the density

V (x − xk ) (≤ +∞) .

%γ (x) := exp{−Eγ (x)}, x ∈

d

,


we have %γ ∈ C 2 (%γ > 0) and

19

∞

X ∇%γ ∇V (x − xk ) . (x) = − %γ k=1

The assumptions (H1) and (H2) are obviously satisfied, but the assumption (H3) needs more delicate considerations. Namely, to show the L1 − uniqueness (cf. Remark 3.7(i)) we still can apply Theorem 4 from [Li99], but the verification of the conditions of this theorem uses the a priori information (6.6) about the given configuration γ ∈ U in an essential way. For the technical details we refer the reader to [KKR02]. Therefore, for all γ ∈ U we can apply all results of this paper to the equation (6.5) and construct a conservative diffusion on the state space d \ γ which gives us the unique solution to the martingale problem associated with (6.5). The main problem remaining open in this model is related to the study of the asymptotic behavior of the particle for t → ∞. In this direction we only have some conjectures based on physical intuition rather than on rigorous results. For an excellent discussion of the physical point of view concerning diffusions with a random velocity field we refer to [BG90] and the references therein. Rigorous mathematical results have so far only been obtained in the case of drifts which are much more regular than those considered above, see e.g. [Ol94].

Acknowledgement The last named author would like to thank the Scuola Normale Superiore, in particular, his host Giuseppe Da Prato for a very pleasant stay in Pisa during which a part of this work was done. Financial support of the SNS di Pisa as well as of the DFG-Forschergruppe “Spectral Analysis, Asymptotic Distributions, and Stochastic Dynamics” is gratefully acknowledged. References [AFKS81] S.Albeverio, M.Fukushima, W.Karwowski, L.Streit, Capacity and quantum mechanical tunneling, Comm. Math. Phys. 81 (1981), 501-513 [AH-KS77] S.Albeverio, R.Hoegh Krohn, L.Streit, Energy forms, Hamiltonians, and distorted Brownian paths, J. Math. Phys. 18 (1977), 907-917 ¨ckner, On partial integration in infinite dimensional [AKR90] S.Albeverio, S.Kusuoka, M.Ro space and applications to Dirichlet forms, J. London Math. Soc. 42 (1990), 122-136 ¨ckner, Differential geometry of Poisson spaces, [AKR96a] S.Albeverio, Y.Kondratiev, M.Ro C. R. Acad. Sci. Paris, Seri´e I 323 (1996), 1129-1134 ¨ckner, Canonical Dirichlet operator and dis[AKR96b] S.Albeverio, Y.Kondratiev, M.Ro torted Brownian motion on Poisson spaces, C. R. Acad. Sci. Paris, Seri´e I 323 (1996), 11791184 ¨ckner, Analysis and geometry on configuration [AKR98a] S.Albeverio, Y.Kondratiev, M.Ro spaces, J. Funct. Anal. 154 (1998), 444-500 ¨ckner, Analysis and geometry on configuration [AKR98b] S.Albeverio, Y.Kondratiev, M.Ro spaces. The Gibbsian case, J. Funct. Anal. 157 (1998), 242-291 ¨ckner, Gradient diffusions with singular drifts in [AKR02] S.Albeverio, Y.Kondratiev, M.Ro models of Mathematical Physics, in preparation (2002) [AKS86] S.Albeverio, S.Kusuoka, L.Streit, Convergence of Dirichlet forms and associated Schr¨ odinger operators, J. Funct. Anal. 68 (1986), 130-148 ¨ckner, Classical Dirichlet forms on topological vector spaces: con[AR89] S. Albeverio, M.Ro struction of an associated diffusion process, Prob. Th. Rel. Fields 83 (1989), 405-434 ¨ckner, Dirichlet forms on topological vector spaces: closability and [AR90] S. Albeverio, M.Ro a Cameron-Martin formula, J. Funct. Anal. 88 (1990), 395-436

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