2 University of North Carolina, Mathematic Department, Charlotte, N.C. 28223-9998, USA. 3 Institute of .... identified with the centers of Voronoi cells forming the corresponding large ...... We call x (k) record points and t/(k) record values.
Probab. Theory Relat. Fields 100, 457-484 (1994)
Probability Theory
o" Related Fields
9 Spfinger-Verlag 1994
Stratified structure of the Universe and Burgers' equation - a probabilistic approach Sergio Albeverio 1, Stanislav A. Molchanov 2, Donatas Surgailis3 1 Fakult~itfiir Mathematik, Ruhr-UniversitfitBochum, D-44801 Boehum, Germany (SFB 237, BiBoS; CERFIM (Locarno)) 2 University of North Carolina, Mathematic Department, Charlotte, N.C. 28223-9998, USA 3 Institute of Mathematics and Informaties, 2600 Vilnius, Lithuania Received: 6 September 1993/In revised form: 20 May 1994
Summary. The model of the potential turbulence described by the 3-dimensional Burgers' equation with random initial data was developped by Zeldovich and Shandarin, in order to explain the existing Large Scale Structure of the Universe. Most of the recent probabilistic investigations of large time asymptotics of the solution deal with the central limit type results (the "Gaussian scenario"), under suitable moment assumptions on the initial velocity field. These results and some open questions are discussed in Sect. 2, where we concentrate on the Gaussian model and the shot-noise model. In Sect. 3 we construct a probabilistic model of strong initial fluctuations (a zero-range shot-noise field with "high" amplitudes) which reveals an intermittent large time behaviour, with the velocity F(t,x) determined by the position of the largest initial fluctuation (discounted by the heat kernel 9(t,x, 9 )) in a neighborhood of x. The asymptotics of such local maximum as t --+ oe can be analyzed with the help of the theory of records (Sect. 4). Finally, in Sect. 5 we introduce a global definition of a point process of t-local maxima, and show the weak convergence of the suitably rescaled process to a non-trivial limit as t ~ e~.
Mathematics Subject Classification (1991):
60F05, 60H25, 60G70, 85A40
1 In~oducfion Modem astrophysical data provide clear evidence that matter in the Universe is highly non-uniformly distributed. The non-uniformity can be observed at very different scales, starting with the size of planetary systems and ending with scales comparable to the observed Universe itself (roughly 1/100 of the latter). The reason for it lies in the gravitational instability which shows up in different forms at different stages of the evolution. A rather good qualitative and partly quantitative description of the stratification of the matter in the "dust cloud" after its particles had already formed and interacted by gravitational
458
S. Albeverio et al.
forces only, is provided by stochastic models based on the Newtonian diffusion theory (Albeverio et al. [ABHK1-4], [ABHKM]). The role of the "cloud" can be played there by protoplanetarian nebulae [ABHK1,2], galaxies [ABHKM], or clusters of galaxies. However one can hardly expect that the above models could explain the recently discovered Large Scale Structure of the Universe. This phenomenon is actively discussed in the modern literature, see e.g. the reviews [ShZ], [ShDZ], [ZMSh], [ZMRS], and the numerous references therein. Indeed, the very scale of the stratum indicates that it was formed very early when the Universe was very hot and dense, which makes a hydrodynamic approach more adequate. The corresponding theory was developped, at the physical level of rigor, by Zeldovich and his co-authors ([Z], [ZMSh], [ArShZ], [ShDZ], [ShZ]). According to it, the evolution of the velocity field g = g(t,x),(t,x) E [0,co) x IR3, of matter is described by the Burgers' equation
{ a~/~t + (~, V)~' = (1.1)
~c3~,
curl F = 0, b~(0,x) = -2~cV~(x),
or the more general equation (1.2)
{
~/~t + (~, V ) ~ = ~A~'- 2 ~ V ~ , curl g = 0, ~'(0,x) = -2~cV~(x),
while the density of matter p = p ( t , x ) , ( t , x ) C [0, oc) x IR3, satisfies the continuity equation: (1.3)
f (~p/~t + div(p~ = 0,
p(O,x) = po(x).
In (1.1) and (1.2), the potentiality condition curl ff = 0 reflects self-gravitation of the medium, and the viscosity parameter ~c > 0 its hydrodynamic friction, i.e. the effect of numerous collisions of particles. The initial fields {~(x): x 6 IR3}, {p0(x) :x E •3} and the external potential {~(x) :x E IR3} in (1.1)(1.3) are assumed to be smooth real stationary random fields. Unfortunately there is no clarity concerning the important question about the type of the distribution of these random fields; one can only hope that the asymptotic behaviour of ~(t,x) and p(t,x) as t ~ oc does not depend much on concrete details of the stochastic model. Both Eqs. (1.1) and (1.2) admit an "explicit" solution via the Hopf substitution ff(t,x) = -2~cVlog u ( t , x ) , where the real function u(t,x) satisfies the linear parabolic equation { ?u/Ot = ~cAu - Cbu , u(O,x) = e ~(x) , which for 9 = 0 reduces to the heat equation
Burgers' equation - a probabilistic approach (1.4)
459
f 8u/Ot = ~ A u , ~. u(O,x) = e ~(x) .
(A detailed discussion of the Hopf substitution including a more difficult case of time-dependent coefficients, can be found in [ABHK2]). Hence the solution of Eq. (1.1) can be written as
(1.5)
f ~7Zg(t, x, y)e~(Y)dy ff(t,x) = R3 f~3 g( t,x, y )e~(Y)d y '
where g(t,x, y ) is the Ganssian kernel g(t,x, y ) ~ (4n~t) -3/2 exp{-]x - y]2/4tct} . In the present paper we discuss the asymptotics of the distribution of b~(t,x), as given by (1.5), as t ~ o~. (The situation is more complex for the solution of Eq. (1.2), where localization effects should be taken into account; however, see [Sil,2], [HLDUZh] for recent developments.) At the present time, there are several models ("scenarios") of the evolution which are basically determined by the magnitude of the initial fluctuations ~(x). In Sect. 2 we discuss the "Gaussian scenario", realized by a version of the central limit theorem, which takes place when the initial fluctuations are weak in the sense that the corresponding exponential moments of ~(x) exist. The discussion is based on the recent paper [BUM] and focuses on two basic models of {~(x)}, namely, the Gaussian model, or the random field {~(x) : x E 1R3} being a stationary Gaussian field, and the shot-noise model." (1.6)
{(x) = ~{i~o((x - xi)/Zi) , i
where cp(x) is a smooth compact function (the "unit potential"), {xi} is a homogeneous Poisson ensemble, and {{i}, {gi} are i.i.d, random variables ("amplitudes" and "scales"), independent of {xi}. Very recently, further papers and preprints appeared on related models ([Bu], [GiMSu], [LEO], [FSuWo], [SuWol,2] ), where in some cases non-Gaussian limit distributions appear as well. The main results of the paper refer to the case of strong initial fluctuations (Sect. 3-5), where we describe the intermittency effect and construct a probabilistic model for the origin mad the evolution of the stratified structure. The model is a degenerate version of the shot-noise (1.6) corresponding to Xi - 1 and zero-range potential q)( 9 ), which in view of the nonlinearity of the initial condition in (1.4), can be rigorously defined by er
= ~er
-
xi).
i
The strength of fluctuations of ~(x), or e ~(x), is reflected in the assumption that the tail distribution function (1.7)
g ( a ) = P[e r > a]
varies slowly as a ---+oc, i.e. for each c > 0 i
460
S. AIbeverio et al. lim H ( a c ) / H ( a ) = 1. a ---~- O O
Under this condition, the limit behaviour of the sum u(t,x) = y'~ig(t,x, xi)er differs very much from the usual Gaussian behaviour in the case when the second moment (e 2~i) < oc, and is determined by its maximal term g(t,x, xi.)e ~i* = maxg(t,x, xi)e ~i, i.e. for each x 6 IR3
u(t,x) - 1, t--.~ g(t,x, xi* )e li*
P lim
see L e m m a 1, also [Da], [MaRe] for related results. According to Theorem 7, under some additional conditions on H(a), for each x E IR 3, (1.8)
g(t,x) ~ x - x i , t
(t --+ oe).
According to (1.8), the structure of the velocity field g(t,x) can be pictured as follows: there are points xi, where the initial field e r = er = xi, takes high values ("local maxima"), from which the velocity ~'(t,x) is pointed outwards, and which remain stable for some time and "serve" simultaneously all points x in their neighborhood. In other words, matter is repulsed from xi. and should concentrate near the boundaries of "Voronoi cells" with centers at these points. After some time, the corresponding "large scale structure" is destroyed, as the local maxima are replaced by higher local maxima, see the definition of xi., resulting in larger cell structure, and so on. The above interpretation is in part speculative and cannot be rigorously established on the basis of Theorem 7 alone, which refers to the asymptotics at a fixed point x. Even in this context, the behaviour of the right hand side of (1.8) requires further analysis. For this purpose, we use considerations based on the theory of records (see [R~], and the review paper [N]) (Sect. 4). Fix x = 0, say, and define ~(k) = ~i(k),X(k) = Xi(k) as the k-th record value and record point, respectively, k = 1,2,..., i.e. i (k+l) = min{i > k
:
~i
>
~i(k)}
,
and we assume that the points xi of the Poisson ensemble are indexed in the concentric order around the origin (see Sect. 4 for details). Then xi. = x (k*) for some k* = k*(t) (Lemma 2), i.e. the corresponding local maximum is attained at a record point. Using a rather standard argument of the theory of records, one can estimate the growth of the record sequence {[x(k)l, ~(k)} and consequently the growth of Ix(k*(t))[ = [Xi*(t)[ as t ~ oo, under some additional conditions on the distribution function (1.7). Roughly speaking, if H(a) decays as (log a) -1/~ with some 0 < 0 < 2/3, then [xi*(t)[ is of the order t 1/(2-3~ up to a slowly varying factor (Theorem 8). Finally, in Sect. 5 an attempt is made to rigorously introduce a global (Voronoi) cell structure in the whole space IR3 related to the asymptotics (1.8), and to study its scaling limit as t ---+ oo. A point xi from the Poisson ensemble will be called a t-local maximum (point) if
er
0, 0) > sup e~Jg(t, xi,xj). j+i
Burgers' equation - a probabilistic approach
46l
Let {x}t)} be the totality of all t-local maxima. In view of (1.8), they can be identified with the centers of Voronoi cells forming the corresponding large scale structure. The main result of Sect. 5 is Theorem 9, which gives sufficient conditions on the distribution function H~(u) = P[~i > u](= H(eU)) for the existence of a non-trivial weak limit {x}~)} of the rescaled point process {btlx}t)}. The normalizing constants bt give the typical scale of the Voronoi cell, b t 3 being proportional to their density. The main condition (5.1) of Theorem 9 requires that He( 9 ) belongs to the domain of attraction of a max-stable distribution and is more specific than the corresponding assumptions in Theorem 8, although their results can be roughly compared and agree in the main parameter 0 E [0, 2/3), see Remark 5.
2 The Gaussian scenario
In this Sect. we discuss the central limit and related problems for the solution (1.5) of the Burgers' equation (1.1), for two models of the fluctuation field {~(x) : x E IR 3} popular in the theory of disordered systems, namely the Gaussian model and the shot-noise model.
2.1 The Gaussian model Let { ~ ( x ) : x E ~x3} be a real stationary Gaussian field with mean zero, i.e. (~(x)) = 0, and covariance B(x - y) = (~(x)~(y)) -+ 0(Ix - y] ---+ oc). Because of the potentiality of ~'(0,x), the zero mean assumption (~(x)) = 0 does not restrict the generality. To guarantee the smoothness (i.e. the existence of continuous partial derivatives) of 4(') a.s., we shall assume the existence and continuity at x = 0 of partial derivatives of the correlator B(x). We shall also assume that B(x) varies regularly at infinity, i.e. (2.1)
B(x) = Ixl-=t(Ixl)Z(x/Ixl),
for some e > 0, and some slowly varying function L( 9 ), with S( 9 ) a strictly positive continuous function on the unit sphere $2 C IR3; in the isotropic case of course E( 9 ) = 1. Set (2.2)
u(t,x) = f g(t,x,y)e~(Y)dy,
(t,x)
E
IN+
X IN 3 .
N3
One can easily compute all moments of the random field {u(t,x) : (t,x) E ~+ x IR 3}; in particular, (2.3)
(u(t,x)) = f g(t,x, y)(e~-(Y))dy = e e(~ N3
and
Cov(u(h,xl ), u ( t 2 , x 2 ) )
= (u(h,x2)u(t2,x2))
- e ~(~
462 (2.4)
S. Albeverio et al.
= f f g(t;,xl, Yl)g(t2,x2, y2)eB(~
B(yl-y2) -- 1]dyldy2
~3 R3
= e ~(~ f g(q + t2, O, xl - x2 - z)[e B(z) - 1]dz ~3
= eB(~
+ t2)) -3/2 f e -Ix1 -x2-zl2/4rC(tl+t2)[e B(z) -- 1]dz. ~3
We shall distinguish between two cases: a > 3 and 0 < ct < 3. In the first case,
(2.5)
f IB(z)ldz < ~ , IR3
which means that the Gaussian field {r : x E 11t3 } is weakly dependent in the sense of asymptotical analysis o f functionals o f Gaussian processes, see e.g. [MM], [DoMa], [Ma]. The case 0 < c~ < 3 corresponds to strongly dependent Gaussian field. The intermediate case c~ = 3 is more delicate and will not be discussed below. In the case e > 3, from (2.3) and (2.4) by the dominated convergence theorem one easily has for the renormalized field U ( t , x ) : = t3/4(u(t, x t l / 2 ) eB(O)/2), (U(t,x)} = 0, that lim (U(t, xa)U(t, x2))
g---+O~
= elg(2,XhX2)
,
where
cl = e 8(~ f (e B(z) - 1)dz > 0 IR3
because of the inequality f~.3(e B(z) - 1)dz > l fm3B2(z)dz, which follows from positive definiteness of Bn( 9 ),n = 1,2,... For each compact set D C IR3, let C(D) denote the Banach space of all continuous real functions on D, with the sup-norm. Clearly u(t,x) and U(t,x) are a.s. continuous in x for each t > 0 and therefore define random elements with values in C(D). Theorem 1 Let a > 3. Then the distribution of the random fieM {U(t,x) : x E IR3} converges weakly in C(D) as t --~ oo, for any compact D C IR 3, to
the distribution of the isotropic stationary Gaussian random field with zero mean and covariance c l g ( 2 , X l , X 2 ) = c1(87ztC)-3/2e-lXl-X212/8r~,xi,xz C ]R 3. The proof o f this theorem is rather standard. In particular, the asymptotical normality o f finite dimensional distributions can be proved by the method o f moments ([BrMa], [SuWol]), in which case the asymptotics (2.1) can be replaced by the integrability condition (2.5) alone. In a similar way, one can consider the joint space-time asymptotics of u(t,x). In the sequel, the convergence of random fields indexed by (r,x) ~ IR+ x IR3 is understood as (weak) convergence o f their finite dimensional distributions, although apparently in many cases functional limit theorems can be proved as well. Theorem 2 Under the conditions o f Theorem 1, the random field {U(zt, x) : (v,x) E IR+ x IR3 } tends as t ~ c~ to the distribution o f the Gaussian fieM
~cl/2w(z,x) : (r,x) E IR+ • IR3}, with zero mean and covarianee
Burgers' equation - a probabilistic approach
463
(w(zl,xl)W(g2,X2)) = ~('t'l -~- "C2,XI,X2).
(2.6)
Let us note that the limiting Gaussian field w(z,x) is infinitely differentiable in (z,x) E IR+ • IR3 a.s. and admits the white noise representation
w(z,x) = f 9(z,x, y ) ~ ( d y ) , ~3
(2.7)
where ~(. ) is the Gaussian white noise in ]R3 with ( ( ( d y ) ) = 0 and
(~(dx)~(dy)) = a(x - y)dx. The asymptotic distribution of the velocity field g(t,x) is obtained from Theorem 2 using the representation
g(t,x) = -2~: Vu(t'x) u(t,x) '
(2.8)
see (1.5), where the denominator tends to a constant ( = (u(t,x)) = e ~(~ by the law of large numbers, while in the numerator the gradient commutes with the passage to the limit. Theorem 3 Under the same conditions as in Theorem 2, the random field
{tS/gv(zt, xtl/Z):(z,x) E ~-+ X ]R 3}
(2.9)
tends as t ---+ cx~ to the Gaussian field {--e 21/2VW('C,X)'('C,X) E JR+ x JR3}, where c2 = c1(21r -B(~ and g w ( z , x ) = f Vg(t,x, y ) ( ( d y ) . ~3
(2.10)
What can be said about the asymptotic distribution of the density of matter, whose evolution is governed by the continuity equation (1.3)? Although we do not any rigorous results, the heuristic argument below shows that it should be Gaussian if the velocity is asymptotically Gaussian, under fairly general conditions on the initial density field {p0(x) : x c 1R3}. From physical argument one can expect that the average density fi = (p(t,x)) = (po(x)) is independent of t and x. Consider the renormalized density fluctuation field
pt(z,x) := t3/4(p(zt, xt I/2) -/~),
(z,x)
E
JR+ x ]R3 .
Then from'(1.3) one has that Pt satisfies the equation (2.11)
Opt/Oz -k div(pYt)t -3/4 + p divot = 0
and the initial condition pt(O,x)=t3/4(po(xtl/Z)-~) , where tV4y(zt, xtl/2), (z,x) c ]R+ x IR3. According to Theorem 3, (2.12)
fft(.C,X)
"-+ C21/2~7W(qT,X)
and, similarly, (2.13)
div Yt(z,x) - - + - c :1/2Aw(z,x)
~t(z,x)=
464
S. Albeverio et al.
as t ---+e~, the convergence being that of finite dimensional distributions, where
w(z,x) is the smooth Gaussian field of (2.7). Passing formally to the limit as t ~ oo in eq. (2.11), one obtains for the limiting fluctuation field poo(z,x) the equation
~p~/Oz = c 21/2p- A w .
(2.14)
The right hand side of (2.14) grows to infinity in the square mean as z ~ 0 and vanishes as z -+ c~ which follows from
((Aw(z,x)) 2) = f (Ay(%x, y))2dy = const, t -7/4 . ~3
This suggests that the limiting field po~('c,x) is independent of po(x) except for the average density/5, and can be written as the Gaussian integral 1/2 -
O(3
po~(%x) = - e 2 p f Aw(s,x)ds 1/2-
= C2 p K
--1
~ l / 2 r . . --1
W(Z,X) = c 2 pm
fg(%x,y)~(dy) N3
An interesting open problem is to make this heuristic argument rigorous. From the astrophysical point of view, the above model does not give interesting asymptotic results: the density of matter becomes uniform (although the convergence is rather slow: p ( t , x ) = ~ + O(t-~/4)); and the motion stops (~(t,x) = 0(t-5/4)); the fluctuations of p and ffhave large scales (0(tl/2)) and are asymptotically Gaussian. In the case where the covariance B(x) of the Gaussian field ~(x) is not absolutely integrable and satisfies (2.1) with 0 < ~ < 3, the limiting distribution of u(t,x) and ~(t,x) can be obtained using the ideas of Dobrushin and Major [DoMa]. According to [DoMa], the limit law is determined by the Hermite rank of f ( x ) = e ~(~ - e 8(~ i.e. the index k = kf of the first non-zero coefficient in the expansion OO
OC
f ( x ) = y~bjHs(x ) = ~ bsHj(x) j=0 j=kf
(2.15)
in Hermite polynomials Hs(x), j = 0, 1,..., and it is non-Gaussian if kf > 2 and eckf < 3. But in our case, oo
(2.16)
eXB(0),/2 = e~(0)/2~B(0)J/2Hj(x)/k!, j=0
so that k f =
the limit distribution of V u ( t , x ) = f ~ 3 V g ( t , x , y ) and of g(t,x) is again Gaussian, although the normalization grows slower than in Theorem 3. In the following theorem the concept of self-similar generalized Gaussian field appears, whose detailed discussion can be found in [Do]. 1 and
x f(r
Theorem 4 Let (2.1) hold with 0 < c~ < 3. Then the random field
Burgers' equation - a probabilistic approach (2.17)
465
{L(tl/2)-l/2t(~+2)/4~(zt, xt 1/2) : (z,x) C JR+ • IR3}
tends as t ~ c~ to the Gaussian field -f
Vg(z,x, y ) ( ( y ) d y : (z,x) E IR+ • ~x3 } ,
IR3 where ( ( ( y ) : y E ~3} is a stationary self-similar generalized Gaussian field with zero mean and covariance (((0)((x)) = ~(x/]x])]xl -~. Remark 1 In the Gaussian model with non-integrable oscillating correlations
B(x) = L(Ixl)lxl -~ cos(x,~)
(0,,~ ~ e3)
for 0 < ~ < 3/2 the limit law of u(t,x) and b*(t,x) is non-Gaussian, given by a second order Ito-Wiener integral, see [SuWol], also [Rol]. The reason for this lies in the fact that because of the oscillation of the covariance, the contribution of the first (or linear) term in the Hermite expansion (2.16) to the limit distribution is smaller than that of the second (quadratic) term. Similar limit distribution for the velocity field was obtained in [LEO] in the case when the initial field ~(x) is the square of a zero-mean Gaussian field. The last result can be easily explained in the framework of the Dobrushin-Major theory, as the Hermite rank of the function f ( x ) = e ~2 -(27z)-l/2f~eax2-xZ/Zdx, O < a < 1/4, equals 2. Finally, we mention the results of [SuWol] on anisotropie Gaussian models with spectral density having singularity along some lines in the frequency domain, in which case the limit distribution of the velocity again is Gaussian, although different from above.
Remark 2 In the important physical works [GSal], [GSa2] (see also [GMaSa]) an attempt was made to obtain non-trivial asymptotic results in the Gaussian framework; however, their works only discusses the 1-dimensional Burgers' equation and is not quite rigorous. {~(x) : x E IR3} discussed above had no intrinsic scale or, more exactly, we ignored its scale. Consider now the following situation when the fluctuation field in the exponent is the sum of a Gaussian field ~(x) (which will be indexed by the lattice Z 3, for simplicity), and an independent random field Aq(x),x E l;E3, of i.i.d, random variables q(x), where A and l are large constants (l E N). Physically it means that we have introduced a large intrinsic scale t, or generated powerful low frequency oscillations, with the frequency v = cl -I for some constant c > 0. Put q(y) = 0 for y r 17Z3 and consider the sum ut(x) = ~ g(t,x, y)e ~(y)+A"(y) ,
y@713
which is a discretized version of u(t,x). Then if Ix] ~ 1/2 and v / ~ l, the "solution" ut(x) ignores "peaks" Aq(y), y E 1773. On the other hand, if Ix] ~ const. (i.e., x remains in the neighborhood of the peak eA~(~ the influence of the Gaussian field {~(y) : y E ~3} is rather weak. Transition phenomena occur at l ~ x/-~, or t ~ 12/tc. At large times t >>12/tr the averaging predominates which
466
S. Albeverio et al.
should produce a similar situation as in Theorems 1 and 2. A serious open problem is to find an appropriate rigorous way to introduce such multiscales, and to investigate the resulting intermittency effects in the spirit of [GSal], [GSa2].
2.2 The shot-noise model This model is more universal than the Gaussian one, and contains several functional parameters: (2.18)
~(x)
= ~iq)((X i
--Zi)/)~i),
X E ]R 3 .
Here, (o(x),x E IN3, is a rapidly decreasing test function (the "potential of unit non-homogeneity"), {~/} is a collection of "amplitudes" and {X/} a collection of "scales", finally, {zi} is a Poisson ensemble in IR3 with intensity 2 > 0. We assume that the random vectors {(~i, Zi)} are i.i.d, and independent of the Poisson ensemble {z/}. Then under mild conditions on the distribution of a generic pair (~,;~), the series (2.18) converges a.s. and in Ll(f2), for each x E IR3, and defines a strictly stationary a.s. differentiable random field whose multivariate characteristic function is given by (2.19)
(exp{in~=taj~(xJ)})=exp{2flexp{i~=laJ~~
} ,
xj E IR3,aj ~ IR,j = 1,...,n (see e.g. [GiSu], [GiSuM], and the references herein). From (2.19) one can compute explicitly various moments of ~(x) and e ~(x) (whenever they exist); in particular,
m = (e ~(x)) = exp / 2 f (Z3 (e~-~~ u )
1))du} ,
Cov(e ~(~ e r ) = m2[e~(x} - 1], where /}(x) = 2 f ((e ~(~/z) - 1)(e r176 IR3
- 1))du.
The magnitude of the fluctuations of the random field ~(x) given by (2.18) depends on the form of qo( 9 ) and the tails of the distribution of (4, Z)- In the case when the fluctuations are "low" (in the sense that the corresponding exponential moments exist), one can prove similar results as for the Gaussian model (although the proof is technically more complicated). Let us formulate a typical result. Theorem 5 [BUM] Let for the shot-noise model ~(x) of(2.18), the following
conditions be satisfied: I~(~)l < c ( 1 +
Ixl) -~
(3~ > 3)
Burgers' equation - a probabilistic approach
467
and
- 1]dx > 0, the statement o f Theorem 3 is true, with c2 replaced by c2 -- (2~:)2f~3 [e&x) - 1]dx. The main idea of the proof of Theorem 5 is rather simple although somewhat cumbersome computationally. Namely, one truncates ~0( 9 ) and Zi by introducing OR(x) = ~p(x) if [xl < R, qgR(X) = 0 if Ix] > R, and Zi,R' = )~i A R ' , R , R ' being large constants. The corresponding shot-noise field
~R,,(x) = ~;~oR((x - z , ) / z ~ , ) ,
x ~ ~3,
i
has finite radius of dependence, and standard probabilistic techniques can be applied. Moreover, some estimates are needed in order to control the passage to the limit as R,R' --+ oo. Theorem 4 also has its "shot-noise" version, where we assume that the potential q~( 9 ) is of the form (2.21)
q)(x) = Ixl-(~+3)/2Lo([xl)Eo(x/[xl),
where 0 < c~ < 3,L0( 9 ) is a slowly varying function, and Z0( 9 ) a strictly positive continuous function on $2. Then in order for the random field ~(x) to be well-defined, in addition to the existence of suitable moments, one has to assume that the conditional expectation (2.22)
({I;g> = 0 ,
see [GiSu], [GiMSu], which implies (~(x)> = 0. Put L = L 2 and
~(X)
= e(r
3)
f Zo(y/lyl)~o((x
+
y)/lx + yl)(lyl Ix + y l ) - a - 3 d y
9
IR.3
Theorem 6 [GiMSu] Let the shot-noise fieM {(x) satisfy conditions (2.20)(2.22), where 0 < ~ < 3. Then the statement o f Theorem 4 is true. Instead of the Dobrushin-Major theory ([DoMa]), which is not applicable in the non-Gaussian case, the proof of Theorem 6 uses a direct approximation of the exponential field e ~(x) - (e ~(x)) in the numerator of (2.8) by the linear field m { ( x ) , m = (er using the fact that (2.19) enables to make an explicit computation of the covariance of the "error" e ~(~) - m m{(x) (see [GiMSu] for details). The above results can be generalized to the situation when the constant intensity 2 > 0 of the Poisson ensemble {zi} in (2.18) is replaced either by a periodic function 2(x) => 0 ([Bu]), or a random field 2(x; co) >- 0 independent of the Poisson ensemble and other random parameters ([FSuWo], [SuWo2]) (in the latter case, the corresponding point process {zi} is called a doubly stochastic Poisson process, or a Cox process, see e.g. [Gr]. In the case of Cox process, non-Gaussian limit laws can appear even if the potential q)( 9 ) has compact support and the moment condition (2.20) is satisfied, provided the random intensity 2( 9 ; co) has long range dependence ([FSuWo]); for example,
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when 2(x; co) is the square of a Gaussian field with slowly decaying correlation function. Finally, one can replace the Poisson or Cox ensemble by a Gibbs ensemble or Gibbs-Cox ensemble (i.e., a Gibbs ensemble with random activity) ([FSuWo]), under suitable conditions on the pair potential and activity, using the techniques of cluster expansions and correlation functions developped in [Ru], [MM] and other works. Although the moment condition (2.20) in Theorem 5 can be weakened (see [BUM], [GiMSu]), the existence of exponential moments of ~ is necessary for e ~(~) to have finite second moment, while some lesser order moments of X are necessary to keep sufficiently low the contribution from distant Poisson centers zi. However, from the physical point of view, it is very natural to consider random fields ~(x) > 0 such that (~k(x)) < cc Vk E N+, or ( e a ~ ) < oo VaEIR+, but (e~(x))=oo V a E N + . As we shall see below, the study of the asymptotics of the solution of the Burgers' equation corresponding to such "high" initial fluctuations ~(x) can help to understand the phenomena of intermittency and formation of large scale structures discussed in the Introduction.
3 Intermittent structures
Let us first describe our model, which is a degenerate version of shot-noise. Namely, we assume that the unit potential has zero range, i.e. there is no interaction between the potentials at different points xi of the Poisson ensemble. Because of nonlinearity of the Burgers' equation, the simplest way to introduce such a model rigorously is by putting e ~(x) = ~ e ~-~6 ( x -
xi),
i
where {{i} is an i.i.d, sequence independent of the Poisson ensemble {x~}. In other words, formula (1.5) for g(t,x) becomes (3.1)
~iVg(t, x, xi)e ~'i g(t,x) = -2,~ ~ .
The function ~(t,x) so defined satisfies the Burgers' equation except for the initial condition; however our main concem is to study large time asymptotics of the solution and its dependence on the distribution of the "amplitudes" ~i. If the distribution of ~i is such that (e Gi) and (e2~i) are finite, then, as in the previous Section, the law of large numbers and the central limit theorem apply to the denominator and the numerator on the right hand side of (3.1), respectively, and the limiting distribution of ~(t,x) is Gaussian. Similar results but with a stable limit law can be obtained i f P [ e ~i > a] ~ const a-~(a ---, oo), for some 0 < c~ < 2, which implies automatically that (e J~i) < eo for every ~t ~ .
However, as we have mentioned before, we shall be interested in the sequel in the situation when (e ~ ) = ec for any ct > 0. More precisely, we shall assume that the tail distribution function
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H ( a ) = Pie ~ > a]
(3.2)
varies slowly as a ~ oc. A typical example of such H(a) is H(a) = (1 + l o g + a ) - ~ , y > 0. Another example is H ( a ) = exp{-c(log+a)l/~),fl > 1, which corresponds to ~i --- ~ , ~i being exponentially distributed with parameter c > 0. As we shall see below, under this assumption on H(a), the main contribution to the sums u ( t , x ) = ~-]~ig(t,x,xi)er and V u ( t , x ) = ~i~Vg(t,x, xi)er comes from a single summand which is, moreover, the same for both sums. If r/t and ~t are d-dimensional random vectors, t > O, we write rhP~t(t -+ oo) whenever Iv, P
(t
stands for convergence in probability. One can easily verify that the relation is reflexive and transitive; for d = 1 it is equivalent to q t / ( t ~ l .
Theorem 7 Let ~(t,x) be defined by (3.1), where {xi} and (~/} satisfy the
conditions above; moreover, let us assume that f H(elxl2)dx < oc
(3.3)
N3 and Lr(a(log a ) - 1/2) (3.4)
--+1
H(a)
(a --+ co) .
Then for each x E IR 3 one has the asymptotics (3.5)
~'(t,x) • t - l ( x - xi* ),
where xi* = xi*(x,O is the point where the random field #(t,x, xi)e ~i attains its maximum, i.e. (3.6)
g(t, x, xi* )e ~i* = max g(t, x, xi )e ~i
9
The following lemma is a generalization of the classical result by Darling [Da] on the asymptotics of the maximum of i.i.d, random variables with slowly varying tail distribution function. L e m m a 1 Let {(Xi, t]i)} be Poisson ensemble on IR3 x ]R+ with the intensity measure dx x dF, F(a) : - P [ q i =< a],a > O, such that H(a) = 1 - F ( a ) varies slowly as a -+ co. For L > O, put Xi = Xi~L = gL(xi)rh,Xd := suPiX~., (3.7)
1L := ~X,,. = ~9L(xi)th, i
i
where the kernels 9L(x) >-_-O,x E IR3, satisfy the followin9 conditions: (3.8)
f H(gL(x)-l)dx < oo, N3
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(3.9)
sup sup gz(x)
< oo,
L>0xEIR 3
and (3.10)
I{x ~ 1~ 3
"gL(x) >
c}l - ~ ~
(L - ~ ~ ) ,
where c > 0 is some constant, and ] 9 I the Lebesgue measure. Then the sum (3.7) converges a.s. for any L > 0 and
(3.11)
/ r L 1 (L ~ oc).
x;
Proof For simplicity of notation, we assume that the distribution fimction F(a) is continuous. Introduce the Poisson random measure N(A x B) := #{i : (Xi,~i) cA • o n ]R 3 x JR+, then [r =
f
agz(x)N(dx • da)
IR3 x N+
is well-defined (i.e.a.s. finite) if and only if
f
(3.12)
l[agL(x) > 1]dxdf(a) < oc
and (3.13)
f
agL(x)l[agz(x) < 1]dxdF(a) < oo;
see e.g. [Me]. Here, (3.12) coincides with (3.8). Denote by iL the left hand side of (3.13); then it can be rewritten as (3.14)
iL = I H(gL(x) -1)I2I(gr(x) -1)dx , p.3
where
IFH(ua) 1]du
(3.15)
H ( a ) := f
o [ H(a)
is well-defined and monotonically vanishes as a ~ oe ([Da]). As gL(x) -~ > cl for some constant cl > 0 and all x E ]R3,L > 0, see (3.9), so the right hand side of (3.14) does not exceed const f~3H(gL(x)-l)dx < cx~. This proves the a.s. convergence of IL; in particular, X [ < ~ and X [ = maxiX/~. Put IZ :=IL - X [ so that IL/X~ = 1 + I Z / X [ and (3.11) follows from (3.16)
lim P[X[ < K ] = 0
( V K < c~)
and (3.17)
lim sup E
K--~a,O L> 0
=0
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471
Here, with GL := {x E ]R3 : g L ( x ) ~ c } ,
P[X[ < K] < P[rli < K/c or xi • GL for all i] = exp{--g(K/c)lGL[} ---+0 ( t --+ oc) according to (3.10), as H(K/c) > 0. Next, if Xi2z = X ~ , so
p[Is
(3.18)
>e,X~ >K] < "* = i]{xk} ] = e - I E ~ E [ X JlJy~i z' X, . . z,L > K, IL i4:j
= e-lE~
f
M(agr(xi)gL(xj) -~) [I F(agr(xi)gL(Xk)-l)dF(a),
i +j K/gL(xi )
k=~i,j
where m(a) := a -1 foudF(u). Integrating by parts, one has M(a) = H(a)I4(a). As gL(x) -1 > Cl > 0 (see above), so tTI(agL(x) -1) --+ 0 (a ---, co) uniformly in x and L. Consequently for any 5 > 0 there is K > 0 such that (3.19)
M(a gL(X)gL(y) -1) < 6H(a gL(x)gL(y) -1)
for all a > K/gL(x) and all x,y E IR3,L > O. Substituting (3.19) into (3.18) we get
oo
< 6 e - l E ~ f H ( a gL(xi)gr(xj) -I ) 1~ f ( a gL(xi)gL(xk) -1 )dF(a) i 4ej O
k ~=i,j
= 6 e - I ~ P [ X j x > Xi,~ > Yk~,k+i,j] = 6e -1 . i4:j As ~ > 0 is arbitrary, this proves (3.17) and the lemma too.
Proof of Theorem 7 We shall consider x = 0 only and assume, similarly as in the proof o f L e m m a 1, that the distribution of ~1i : = e~i is continuous. Introduce the notation u(t) = F Y ~ = ~ g t ( x i ) ~ i , i
i ---- Egl,t(Xi)V]i, i
V l ( t ) ~-~ j l , i i
where
gt(x) = t3/2g(t, O , x )
,
ffl,t(X ) = -- 2~:t5/2V g( t, O,X) = --Xgt (X) . Put also g~Ax) =
I~Ax)l,
Then Y(t) = b~(t,0) can be written as
~(t) -
~l(t) tu(t)
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and the relation (3.5) becomes
~'(t) • - t-lxi 9 ,
(3.20) which is equivalent to
Ot := ~ 1
(3.21)
V*l(t) u(t)
+xi*
e---~O(t --+ 00).
one can easily see that {r/i}, {gt(x)}, {gl,t(x)} satisfy the conditions of Lemma 1, provided t > (4x) -1. Hence by Lemma 1,
u(t) 5 1
(3.22) and
Ei=~i~Xl'i L 0
(3.23)
Xl,i~
as t --+ c~, where Xl,i T = max,.. Xl,i. N o w , Ot
0, where cl = Cl(H,c) is some constant, which follows from monotonicity and slow variation of H(a). Consequently the right hand side of (3.25) does not exceed ClE[N(6v/7)/N(x/7)] = c163, which proves (3.24). P Now, using the easy observation that ~/i.--+~ (t -+ oo), the lemma follows from lira lim sup P[]xi{[ > K]xi* [, [xi* [ > 6v/7, t/i. > 1] = 0 .
(3.26)
K---+ o ~
t---+ o o
Denoting PK,t the last probability, one can rewrite it as
PK,t = E
~
P[Xp a~,lxj I>Klxd oo
=E
oo
~ f dF(a) f dF(b)l[agl,t(xi)/gl,t(xj) Ixil>aVT,lxjl>Klxil 1 0 < b < agt(xi)/gt(xj)] • rI F((agt(xi)/gt(Xp)) A (bgl,t(xj)/glZXp))) p#i,j o~
a,/7,1xjl>Klxil 1 + r I F(agt(xi)/gt(Xp)), p#ij
where [a]+ = a V 0. Now we shall use for the first time condition (3.4) of the theorem. We claim that for any 6 > 0 and # > 0 there is K < c~ such that the inequality (3.28)
[H(agl,t(x)/gl,t(y)) - H(agt(x)/gt(y))]+ < ptt(agt(x)/gt(y))
holds for all a > 1,x, y E R 3 , [ y [ > K]xl,[x [ > 6x/t,t > 1. Assuming temporarily that (3.28) is true, we have by (3.27) that oo
Pg,t
as
f H(agt(xi)/gt(xj)) 1-I F(agt(xi)/gt(Xp))dF(a) A, then (3.28) is trivial cause of monotonicity of H ( 9 ). Let B < A. We have clearly from a > 1 [y[ > Klxl that A > exp{(K 2 - 1)c52/2tr ---+oc(K ---+ oc), moreover,
B A
not the beand
Ixl
gl,t(x)gt(y) gl,t(y)gt(x)
[Y[
and
lyl 2 - I x l 2 t for some constant Cl. Hence (3.29) is satisfied if the inequality logA >_ cl
Ixl 2 >
t
lyl = =
C27y ? -
[xl 2
holds for x, y, t satisfying the conditions in (3.28), and some constant ce > 0. But this is clear from Ixl2/YI2 - Ixl2 > 6]2 ( txl2"~ > r 7 = 1-lYla j =
-- K -2) = c2 > 0
[yl 2
if K > 1. This proves (3.28) and the theorem too.
4 Intermittency and records As we have mentioned in the Introduction, the result of Theorem 7 differs very much from the corresponding results for the Gaussian scenario described in Sect. 2. According to Theorem 8 below, under rather general conditions on the tail distribution function H ( a ) = Pie ~i > a], the velocity ff(t,x) decays at most as t -1/2 up to a slowly varying factor, which is much slower than the rates given in Theorems 3-6, and may even grow with t. In the study of the asymptotics of the right hand side o f (3.5), or the positions xi*(x,t) of "local maxima", as t --+ co, we use an argument based on the theory o f records, see [R6], [G] and the review paper [N].
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Consider the marked Poisson ensemble {(xi, t/i)} in IR 3 • P,+ discussed in the previous Section. We shall assume that the distribution function F ( a ) = P[t/i < a] is continuous and strictly increasing, and that the ensemble is indexed in the concentric order from the origin, i.e. x1 is the point closest to 0 E lR3,x2 is the point closest to 0 except for xt, etc. (Of course, Ixil ~=Ixjl(i=#j) a.s., by the well-known property of the Poisson distribution.) Put x (I) = xb t/O) = t/l, and X ( k + l ) m_ Xi(k+l),t/(k+l) = t / i ( k + l ) , where
i(k + 1) = min{/" > i(k) : tb- > t/i(k)}, k = 1,2,... We call x (k) record points and t/(k) record values. Usually records are defined for a given i.i.d, sequence and not related to Poisson points, see [R6], [N]. It is clear that the distribution of the sequence {x (k)} of record points is universal, i.e. does not depend on F ( 9 ). From monotonicity of the function e -rz/4~ct,r > 0, and the definition of the record sequence (x (k), t/(k)) it immediately follows Lemma 3 Under the above conditions, for any t > 0 Xi* = X (k*) ,
where k* = k*(t) is the index of the record which maximizes t/(k)g(t,O,x(k)), i.e.
~(k*~ a(t, O,x (~*~) = ma~ t/(k~g( t, O,x(k~) . Consider the special case when t/i are exponentially distributed, i.e. (4.1)
F ( a ) = 1 - e -a,
a > O,
F(a) = 0, a < 0. A remarkable property of the exponential distribution is that the distribution of the record sequence {(x (k), t/(k))} can be explicitly described; in particular, q(k) is a sum of k independent exponentially distributed random variables. Put V (k) = 43~ Ix(k)p, O ~k) = x(k)/Ix(k) I E $2(:= {x E IR3 : Ixl = 1}),k = 1,2 .... Lennna 4 In the case of (4.1), the distribution of {(V (k), t/(~))} coincides with
the distribution of the solution of the difference equation (4.2) (4.3)
V (k+l) = V (k) q- etl(k)gk+ t/(k+l) = t/(k) + Ak+l,
1,
k = O, 1..... v(O) = q(o) = O, where {ek}, {Ak} are mutually independent sequences of independent exponentially distributed random variables; {O (k)} is a sequence of independent random variables distributed uniformly on Sz, and independent of the sequence {(V (k), q(k))}. Lemma 4 can be proved by a direct computation; noting that {(V (k), O {k), q(k))} forms a Markov chain whose transition probabilities can be easily found (see also [R~] and [T] for related results on exponentially distributed i.i.d. sequences indexed by integers).
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Solving eqs. (4.2), (4.3) one obtains (4.4) (4.5)
log V (k) = q(k-1) + log(ek + ek-1 e-Ak-1 Jr-... -~- el e-Ak-1 . . . . . AI ) , t/(k) = A1 + . . . q- Ak 9
The sum under the sign of the logarithm in (4.4) converges in probability to a strictly positive random variable; moreover, one can easily check that log(ek + . . . + ele -Ak-1-'''-A1) = O ( l o g k ) a.s. as k ~ oe. Hence the central limit theorem and the law of large numbers apply to {log V (~)} and {q(k)}. In particular, according to the law of iterated logarithm, for any e > 0 there is k0 = k0(e; co) ~ N + such that for all k > k0 (4.6)
k - (1 + e)(kloglogk) 1/2 < q(k) < k + (1 + e)(kloglogk) 1/2
and (4.7)
ek-(l+e)(k ioglogk) 1/2 < v(k) < ek+(l+e)(k loglog k) 1/2 .
If the distribution function F(a) of t/i is continuous, strictly increasing and F(a) < 1 Va > 0, then qi is a monotone transform of the exponentially dislawr --1~ - a . ~
tributed random variable ~ i ' ~ l i = r l (e ',), where H - l ( 9 ) is the inverse function of H ( 9 ) = 1 - F ( 9 ) and P[71i __< a] = 1 - e -a. Using the universality of record points together with (4.6), (4.7), we obtain L e m m a 5 Let F ( 9 ) satisfy the conditions above. Then there are constants C, c, such that for all sufficiently large k > ko(co) (4.8)
Ce~-c(ki~176
1/2 < Ix(k)[ < cek+c(kl~176
1/2
and (4.9)
< rl(k) < H-l(e-k-c(kl~176
H - l ( e -k+c(kl~176
Theorem 8 Assume that there are 0 < 0 < 2/3, 1/2 < 0 < 1,c > 0, such that for all sufficiently large x > 0 (4.10)
x~ -cO~
< l o g H - l ( x -1) < x~e cO~
.
Then for any 0 ~ > 0 and a.e. co C (2 there exists to = to(O~,co) such that for all t > to (4.11)
tl/(2- 3~ e-(l~176
< Ixi * I < tl/(2- 3~ e(l~176 9
Proof As xi* = x (k*), with k* = k*(t) the corresponding index of the record sequence {x (k), t/(k)}; see L e m m a 3, it suffices to prove (4.11) with xi* replaced by x (k*). Put
477
Burgers' equation - a probabilistic approach II(k) = II(k; t) := log q(k)
Ix(k)[2 4~ct '
3 k+ = k+(t) . - 2 - 3~0 log t + (log t) ~ k_ = k _ ( t ) . - 2 - ~3 30 log t - (log t) ~ Assume for simplicity that k+, k_ are integers. Then (4.1 1) follows from (4.12)
II(k_) > max II(k),
(4.13)
II(k+) > kk+
and (4.14)
Ix(k+)l < tl/(2-30)e(logt)Ot ,
(4.15)
Ix~k->I >
tl/(2-aO)e-(logt)
~' "
Indeed, (4.12) implies k* < k+, or [x~k*>l < Ixl, and hence together with (4.14) the second inequality in (4.1 1); the first one follows from (4.13), (4.15) in a similar way. Choose 0 < 0" < 0', then from (4.8), (4.9), (4.10) one has for all sufficiently large k > k0(0"; 09) exp{Ok-k ~
< II(k) < exp{Ok + k ~ }
3 + k~ - l e"x p ( 2 kt -3k O " )
In particular,
> eOL-k~-" [ 1--exp { -- ( ~ ) --0
(logt) ~
~ }] > 0
while for k > k+,
[1 1-exp
0/k+ }J -0
(logt) ~'+o((logt) ~)
< 0.
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This proves (4.12) and (4.13) is proved similarly. Inequalities (4.14), (4.15) easily follow (4.8) and the definition of k+. Theorem 8 is proved.
Examples. It can be easily seen that the "logarithmic" distribution function (4.16)
F(a)=l-log+
yaV1,
7 > 0,
satisfies the conditions of Theorems 7 and 8 provided 7 > 3/2, with 0 = 7 -1. According to these theorems, the solution b~(t,0) of the Burgers' equation behaves asymptotically as t (3-y)/(2r-3), up to a slowly varying factor, in particular, for 3/2 < 7 < 3 the velocity increases in probability as t + ec. The distribution function (4.17)
F(a) = 1 - e x p { - c ( l o g + a ) l / ~ } ,
c > O, fi > 1,
mentioned in the beginning of Section 4, also satisfies conditions of Theorems 7 and 8. In this case, 0 = 0, and the velocity decays as t -1/2, up to a slowly varying factor, for any c > 0, fl > 1. According to Theorem 8, the transition from xi* = x (k) to x/. = x (k+l) occurs during time _ e(~ -~ - e (2-~ = O(e(Z-~ If k is interpreted as the number of the corresponding "Voronoi structure" (see the Introduction), one lk can say roughly that its cells have the diameter _~ e3 and live time -~ e(~--~ after which they are replaced by larger cells around higher local maxima of the random field {t/i}. Of course, this interpretation and even the notion of "local maxima" is not rigorous and requires further investigation.
5 Asymptotics of local maxima Let {(xi,(i)},(i = logt/i, be the marked Poisson point process on ~ d • IR discussed in Sections 3-4. For simplicity, we assume below that the tail distribution function Hr = P[~i > U](~ H(eU)) is continuous, strictly monotone and strictly positive for all sufficiently large u, and the viscosity parameter = 1/4. Assume that a chosen point x E IR3 is a point of the point process, i.e. x = xi for some i. According to Theorem 7, the velocity g(t,x) at this point is determined by the value of the mark ~i = log qi at some possibly different point xj satisfying the condition: ~jg(t, xi,xj) > tlig(t, xi,xi), or
< 4:- Ixi- xjl2/t, provided such j + i exists. Call (Xi,~i) a t-local maximum if there is no such j, i.e. if for any j +i,
4i >>-~j --[Xi- Xj[2/t. Let {(x} '1, {i(t) )} be the point process consisting of all t-local maxima. Our aim below is to study the weak convergence of a suitably rescaled process of tlocal maxima as t --+ 0% under certain conditions on the distribution function of the mark. Let us introduce some terminology. Let be given an open set X c IRd (d > 1), and JV'(X) be the space of all simple locally finite point measures
Burgers' equation a probabilistic approach
479
N on X, with the topology of vague convergence of Radon measures ([K]). In particular, any measure N E JV'(X) can be identified with a countable set {x~} C X of its atoms, i.e. N(A) = #{i : x~ E A} for any Borel set A C X, and N(A) < oc for any compact A C X. A simple locally finite point process on X is a measurable mapping of some fixed probability space (f2,P) into ~Ar(X). The family of all such point processes N -- {xi} will be denoted by N(X). A family of point processes {x/~} E N(X) weakly converges as 2 --4 0 to a point process {x~ E N(X), denoted by {x/~} ~ {x~} if the corresponding distributions P{x~?}-1 on ~/'(X) weakly converge to the distribution P{x~ -1. It is well-known [K] that {x~} ~ {x/~ if and only if for any f E Co(X), ~ J ( x ~ ' ) converge in distribution as 2 ---* 0 to the random variable ~if(x~ Let be given a point process {(Yi,~i)} E N(IR 3 x IR), and a symmetric Borel function 0 =< qS(x)= r < +oc, x E 11t3. The O-thinned point process {(y(r162 (of the point process {(y~,~)}) is defined as follows: for any i f c c+(~ 3 • ~), E f ( y } ~), ~(r ) : = ~ f ( y ; , ~ ) l [ ~ i
__> ~j - r
- yj)vj+i].
i
In particular, the point process {(x}t), ~It))) of t-local maxima introduced above, coincides with the et-thinned process of the Poisson process {(xi, ~i)}, where
et(x) =
Ixl2/t,
The thinning operation is different from the usual independent thinning (i.e. when each point is deleted independently with some probability p E (0, 1), see e.g. [K]), and does not preserve Poisson distribution. It becomes more ~((r ((r ,, intense as t increases or r 9 ) decreases, i.e. ~ Yi i )) C {(y}r ), ~i(Oft))} if r __< r E ]Rd. Let H~-I( - ) be the inverse function of the tail distribution function He( 9 ), which is well-defined, continuous and strictly monotone on (0, l), H ( ~ ( 0 + ) = +ee, by the assumptions made in the beginning of this Section. Theorem 9 Assume that there exist A(2) = H~-1(1/2) and a function B(2) > 0,2 ~ 1, regularly varying at infinity with exponent 0 E [0,2/3), such that for any u E IR there exists the limit (5.1)
lira 2/-/r
+ uB(2)) = Gr
E [0, + o e ] .
2-4c~
Then there are constants at, bt ~ oc (t --+ oc) and a point process {(x}~176 ~}~))} E N(IR 3 x IR) such that (5.2)
g(x i(~), ~(~)~s {(xi(o /bt,(~i(,) -at)t/b2t)} =~ ~ i ,~ (t --+ o~).
The limit point process ~txi , Q )~ coincides in distribution with the 01thinned process of a Poisson point proeess on 1R3 x (G~, +oo), G~ := inf{u E IR : G~(u) < oc}, having the intensity measure -dxdG~(u).
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R e m a r k 3. Condition (5.1) implies that the distribution function F~(u) = 1 H~(u) belongs to the domain of attraction of a max-stable distribution, see e.g. [G], [LLiRo]. Namely, for any u E IR, lim F~(A(n) + uB(n)) = e -6r
.
?/---+0 -1/c,Gr
+cxz, i f u =< - 1 / c , where c > 0,7 > 0, or G~(u) = e -cu ,
u E R , with c > 0. Moreover, in the first case, H i ( u ) is necessarily regularly varying at infinity with the exponent - y [LLiRo]. Note also that the distribution functions F~(u) = 1 - u+ ~ A 1 and F~(u) = 1 - e -~(u+)l/~, u+ = u V 0, corresponding to the distribution functions (4.16) and (4.17), respectively, satisfy condition (5.1).
P r o o f o f Theorem 9 We shall discuss the case G~- = - ~ only, for simplicity. Put C() 0 := B(),3)/~ 2, then C(2) varies regularly with the exponent 30 - 2 E [ - 2 , 0), C(2) ~ 0(2 ~ ~ ) . Its generalized inverse C(-1)(u) :=- inf{2 E [1, ~ ) " C(2) < u} is defined on (0, C(1)], is monotone increasing to ~ as u ---+ 0, and satisfies (5.3)
lira C ( C ( - 1 ) ( u ) ) / u = 1. u--+0
Moreover, C(-1)(u) varies regularly with exponent 1 / ( 3 0 - 2) as u ~ 0, see [BiGTe], i.e., it can be represented as
C ( - O ( u ) = L(1/u)u-1/(z-3~
,
where L( 9 ) is a slowly varying at infinity function. Put (5.4)
bt := C(-1)(1/t) = L(t)t 1/(2-3~ , at : = A ( b ~ ) ,
and
{(xi,t, ~i,t)} := {(xibt~,(~i - at)tb?2)} . Clearly, {(xi,t, ~i,t)} is a Poisson point process on IR3 • IR, with the intensity measure -dxdG~t)(u), where
G~t)(u) := b~H~(at + ubZ/t) . Notice that (5.5)
{(xi,t, ~i,t)} ~ {(Xi,o, ~i,o)} (t ---+ oG),
Burgers' equation - a probabilistic approach
481
where {(Xi,o, ~i,o)} is a Poisson process on 11t3 x IR with intensity measure -dxdGr (5.5) follows from vague convergence of the corresponding intensity measures, or the relation
t~m G~t)(u)= G~(u)
(5.6) for any u C F,.. Now,
G~O(u) = b~Hr
+ uB(b~)Cl(t)) ,
where C](t) := (tC(C(-1)(1/t))) -1 and l i m t ~ Cl(t) = 1, according to (5.3). The limit function G~(.) being continuous on ( G ~ - , + o c ) = lR, see Remark 3, from monotonicity of He( 9 ) we infer that the convergence (5.1) is uniform on any compact set, hence (5.6) follows from (5.1) and the argument above. This proves (5.5). Now, for any f C C~-(~ 3 x IR), ~-'~le(x!t) Z..dJ% l ' i
(5.7)
=
~}t))
=
v"~ z'r (r162
2 - . , J l , x i ,t i
, gi t
~f(xi,t,~i,t)l[~i,t >= ~j,t -Ix~,t-xj,,I 2 Vj+i]
=:
cP({(xi,t , ~ i , t ) } )
9
i
Note that the functional ~( 9 ) is continuous everywhere o n JV'(]R3 X IR) except maybe on the set D0 := {{(yi, ui)} C JV'(IR 3 x P,) : ui = uj - lYi - yj[2 for some i4=j}. It is clear that for the Poisson process {(Xi,o, ~/,o)), the probability of the set Do is zero. Thus the statement of the theorem follows from (5.5), (5.7) and a well-known result about weak convergence ([B], Th. 5.1).
Remark 4. Using Theorem 9 and well-known properties of Poisson process, one can find n-point correlation functions [Ru] P(~)(Yl,..., Y~) = P [ N ( ~ ) ( d y l ) = 1,..., N(C~)(dyn) = 1]/dyl ...dyn, Yi C 11{3, yi=l=yj(i~j), i,j = 1. . . . . n,n = 1,2 .... of the (unmarked) limit process {x}~)}; N ( ~ ) ( A ) := #{i: x}~) E A}. Namely, p~)(yb...y,)=(-1)
n f U(y)n
exp
/ (
-f
n
VG (uj+lx-yjl2)dx
dGr
R3 i=1
where
U(y)n := {(U)n E ( G ~ , ~ ) n " ui >_ uj - l y i - yjl 2,
i + j , i,j = 1.... ,n} .
Remark 5. If B(2) =< 2~ cffl~176176 for some C 1 > 0, 1/2 < z9 < 1 and all sufficiently large 2 > 0, then condition (5.1) of Theorem 9 implies condition (4.10) of Theorem 8, with the same 0 and tg. The normalizing constant bt can be interpreted as the average distance between t-local maxima, or the scale of a typical Voronoi cell centered at these points. A similar interpretation can be given to the distance [xi* ] in Theorem 7. From Theorems 8 and 9 we have
482
S. Albeverio et al.
the approximate equality: bt ~- [Xi*(t)l ~ t 1/(2-30), up to a slowly varying factor. This suggests a rough agreement between both interpretations and the results of Theorems 8 and 9, at least in the main parameter 0.
Acknowledgements. The financial support of the SFB 237 (Bochum-Essen-Dfisseldorf) and the EC-Project "Stochastic Processes in Analysis" is gratefully acknowledged. We thank Prof. G. Shandarin for stimulating discussions.
References
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