Brownian Motion in Correlated Poissonian Potential

Superdiffusivity for Brownian Motion in a Poissonian Potential with long range correlation I:
Lower bound on the volume exponent

Hubert Lacoin CEREMADE, Place du Maréchal De Lattre De Tassigny 75775 PARIS CEDEX 16 - FRANCE lacoin@ceremade.dauphine.fr
Abstract.

We study trajectories of -dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyper-plane. Our Poissonian potential is constructed from a field of traps whose centers location is given by a Poisson Point process and whose radii are IID distributed with a common distribution that has unbounded support; it has the particularity of having long-range correlation. We focus on the case where the law of the trap radii has power-law decay and prove that superdiffusivity hold under certain condition, and get a lower bound on the volume exponent. Results differ quite much with the one that have been obtained for the model with traps of bounded radii by Wühtrich [20, 21]: the superdiffusivity phenomenon is enhanced by the presence of correlation.
2000 Mathematics Subject Classification: 82D60, 60K37, 82B44
Keywords: Streched Polymer, Quenched Disorder, Superdiffusivity, Brownian Motion, Poissonian Obstacles, Correlation.

Résumé.

Dans cet article, nous étudions les trajectoires d’un mouvement brownien dans évoluant dans un potentiel poissonien jusqu’au temps d’atteinte d’un hyper-plan situé loin de l’origine. Le potentiel poissonien que nous considerons est construit à partir d’un champs de pièges dont les centres sont déterminés par un processus de Poisson et dont les rayons sont des variables aléatoires IID. Nous concentrons notre étude sur le cas particulier ou la loi des rayons des pièges à une queue polynomiale et nous prouvons que les trajectoires ont un caractère surdiffusif quand certaines conditions sont vérifées et nous donnons une borne inférieure pour l’exposant de volume. Les résultats sont sensiblement différents de ceux obtenus dans le cas ou les pièges sont à rayon bornés par Wühtrich [20, 21]: le phénomène de surdiffusivité est renforcé par la présence de corrélations.

1. Introduction

1.1. Brownian Motion and Poissonian Traps

This paper studies a model of Brownian Motion in a random potential. Given a random function defined on and ( being the inverse temperature) we study trajectories of a Brownian motion killed at (space-dependent) rate conditioned to survive up to the hitting time of a distant hyperplane.

The potential that we considered is buildt from a Poisson Point Process on with intensity where is the Lebesgue measure, and is a probability measure. We call it . The definition of is where is a non-negative function with compact support (for the sake of simplicity we restrict ourselves to , where denotes the euclidian ball). The potential can be seen as a superposition of traps centered on the points , and with IID random radii . We are specifically interested in the case where has unbounded support and a tail with power law decay.

This model is very similar to the ones studied in [16, 17, 19, 20, 21] (see also the monograph of Sznitman [18] for a full acquaintance with the subject), the only difference is that we allow the traps to have random radii. The crucial difference is that the potential we consider has long-range spacial correlation i.e. that the value of at two distant point are not independent but have correlation that decays like a power of the distance. Another situation where one has a correlated potential is when is not compactly supported (see e.g. [4, 12]) but this out of our scope.

1.2. Superdiffusivity and volume exponent

A typical trajectory of a Brownian Motion killed with homogeneous rate and conditioned to survive till it hits a distant hyperplane looks like the following:
The motion along the direction that is orthogonal to the hyperplane (call it ) is ballistic (with speed ) but the motion along the other coordinate is diffusive, and for that reason trajectories tend to stay in a tube centered on the axis of diameter where is the distance between the motions starting point and the hyperplane that has to be hit.

Adding a non homogenous term to the killing rate makes the problem much harder to analyze and changes this behavior in some cases: physicists predicts that when is large (at low temperature) or when for every , transversal fluctuation of the trajectories are superdiffusive i.e. of an amplitude for some that is called the volume exponent. The aim of the paper is to show that spatial correlation in enhances that phenomenon.

2. Model and results

2.1. Model

Let us make formal the definition we gave for the model. We consider

(2.1)

a Poisson Point Process in (we index the points in the Poisson Point Process in an arbitrary deterministic way, e.g. such that is an increasing sequence, being the euclidian norm on ) whose intensity is given by where is the Lebesgue measure on and is the probability measure on defined by

(2.2)

for some (which is a parameter of the model). We denote by , and its associated probability law and expectation.

Given , let , be defined as

(2.3)

Note that , for almost every realization of and for every if and only if the condition

(2.4)

is fulfilled, and we always consider it to be so in the sequel. This construction is natural way to get a potential with long range correlation that decays like a power of the distance constructed from a Poisson Point Process. Indeed with this setup,

(2.5)

Given , let (and the associated expectation) denote the law , standard Brownian Motion starting from and set . Given set

(2.6)

Given any closed set let denote the hitting time of . Given , the probability for a Brownian Motion killed with rate to survive till it hits is equal to

(2.7)

The law of the trajectories conditioned to survival is given by

(2.8)

In what follows, we consider only the case as temperature does not play any role in our results.

2.2. Review of known results

Let us turn to a rigorous definition of the volume exponent. For one defines the be a tube of cubic section, of width and centered on , where .

(2.9)

and the event

(2.10)

In words, is the event:“ has transversal fluctuation of amplitude less than ”.

We define the volume exponent as

(2.11)

It is expected to coincide with

(2.12)

In particular if one has .

Let us recall what are the conjecture and known result for the volume exponent for the model of Brownian Motion in Poissonian Obstacles studied in [16, 17, 18, 19, 20, 21] and for related model. In the remainder of this section, relates more to the general notion of volume exponent that to the strict definition given above and in the different results that are cited, definitions may differ:

When whether or not should depend on the temperature i.e. on the value of : at high temperature (low ) trajectories should be diffusive and satisfy invariance principle whereas at low temperature (high ) trajectories are believed to be superdiffusive. In the low temperature phase, the value of should not depend on . Diffusivity at high temperature has been proved for a discrete version of this model by Ioffe and Velenik [5] and it is reasonable to think that their technique can adapt to the Brownian case when correlation have bounded range. Prior to that, similar results had been proved for directed polymer in random environment that can be considered as a simplified version of the model (see e.g. [2, 3]). Superdiffusivity at low-temperature is a much more challenging issue: physicists have no clear prediction for the value of and no mathematical progress towards proving has been made so far.

In any dimension, the value of is conjectured be related to the fluctuation of around its mean: If the variance asymptotically satisfies

(2.13)

then one should have the scaling relation

(2.14)

The heuristic reason for this is that is the entropic cost for moving away from the axis , whereas the energetic gain one might expect for such a move is . The volume exponent corresponds to the value of for which cost and gain are balanced.

When , there is no phase transition and trajectories are expected to be superdiffusive for all . It is not very clear what it means when but for the two-dimensional case, physicists predicts on heuristic ground that and . This is conjectured to hold not only for Brownian Motion in Poissonian Obstacles but for a whole family of two-dimensional models called the KPZ universality class (directed last-passage percolation, first passage percolation, directed polymers…). In fact the conjecture goes much further and includes a description of the the scaling limit (see e.g. the seminal paper of Kardar Parisi and Zhang [7]).

A lot of efforts have been made to bring that conjecture on rigorous ground. In fact, it has even been proved that for some very specific models in the KPZ universality class:

  • Directed last passage percolation in dimension with exponential environment by Johansson [6].

  • Directed polymer in dimension with -Gamma environment and specific boundary condition by Seppalainen [18].

These two results have in common that they have been proved by using exact computation that are specific to the model. Note that a similar result has been proved for the conjectured scaling limit of this model, in [1].

Another approach has been to look for more robust method using the idea of energy vs. entropy competition. In [20, 21] , Wühtrich proved that for and that in all dimension (with a definition for that is slightly differs of the one we present here). In [19], he proved a rigorous version of the scaling identity . Similar results had been proved before for first passage Percolation by Licea, Newman and Piza [10] and after for directed polymers by Peterman [13] and Méjane [11].

In [8], we have investigated the effect of transversal correlation in the environment for directed polymers, and in particular their effect on the volume exponent. There it is shown that in any dimension, if environment correlations decay like a small power of the distance then, superdiffusivity holds. More precisely that if the correlation decays like the inverse-distance to the power , then . In some cases it shows in particular that which indicates that KPZ conjecture does not holds in that case. The bound of [11] remains valid.

Here we study the effect of isotropic correlation (and therefore it seemed natural to to it in for an undirected model), and we have not found in the literature any prediction about what the value of should be.

2.3. Main Result

We present a lower bound bound on for our model with correlation.

Set

(2.15)
Theorem 2.1.

(Lower bound for the volume exponent)

For any choice of , , , one has

(2.16)

where is the quantity defined in (2.11).

Remark 2.2.

In some cases, the lower bound that we get for is larger than , which contrasts with all the results that we have reviewed in the previous section and indicates that isotropic correlation enhance superdiffusivity in a more drastic way than transversal ones. The above result gives a necessary condition for having superdiffusivity: and or and .

Remark 2.3.

The definition of the volume exponent that we use is different of the one used in [20] which is slightly weaker. Combining techniques used in [8] and here one could prove also that when for any value of and for this definition of .

2.4. Further questions

We prove in this paper that for a class of correlated environment, the trajectories have superdiffusive behavior and that the bound that is valid for the uncorrelated model [20] is not valid here and can be beaten. Therefore one would be interested to find an upper bound () for . We have addressed this issue in companion paper [9]. In some special cases (when either ) one can even prove that the lower bound that we prove here is optimal and give the exact value of .

The result that we present concerns the so-called point-to-plane model. A similar result should hold for the point-to-point model. The method that we use in Section 3.3 and 3.4 are quite robust and could be easily adapted to the other setup but getting something similar to what is done in Section 3.2 seems more difficult and challenging and we are not able to do it yet. One can still get a non optimal result by using another construction inspired by what is done in [21], we present it in the Appendix.

For the Brownian directed polymer in correlated environment, in [8], it is shown that either superdiffusivity holds at all temperature or that one has diffusivity at high temperature (except for some special limiting cases) . For the model presented here one would like to show something similar e.g. that diffusivity holds if correlation have fast-decay at infinity (decay like a large power of the inverse-distance) and the amplitude of is small. For the moment this is quite out of reach and the methods used in [5] do not seem to adapt to this case.

3. Proof of Theorem 2.1

3.1. Sketch of proof

In order to make the strategy of the proof clear we need to introduce some notation. One defines

(3.1)

and

(3.2)

where for a closed set , and , denotes the Euclidean distance between and , i.e.

(3.3)

( is the Euclidean norm), and , is the Euclidean ball of radius . Let be the set of trajectories that avoids the set .

(3.4)

Note that, as Brownian trajectories are continuous

(3.5)

The first step of our proof (Section 3.2) is inspired by [10]. We prove a result much weaker that Theorem 2.1 by using a simple geometric argument combined to rotational invariance: that with probability close to one,

(3.6)

or equivalently, that with probability close to one,

(3.7)

where for an event , we use the notation

(3.8)

Then we modify slightly the environment () by adding additional traps whose radii are in , and whose centers are in the region . The second step of the proof (Section 3.3) is to show that typical realization of are roughly the same as typical realization of .

Finally , we notice that adding these traps lowers the value of but that . (adding these traps changes the values taken by only in the region that the trajectory in the event do not visit). The third step of the proof (Section 3.4) is to show that with our choice of and , one has with large probability

(3.9)

for some , which combined with (3.7), gives the result with instead of . The fact that and look typically the same allows to conclude.

We explain in the course of the proof the reasons for our choices of and how we obtain the condition on .

3.2. Using rotational invariance

For , let denote following the rotation of

(3.10)

Set

(3.11)

(the image of the sets resp.  for ). One defines in the same fashion the event as

(3.12)

Note that if if and only if .

One proves the following

Proposition 3.1.

For any set . Then one has that for any (for some fixed small enough ),

(3.13)

In particular, setting one has

(3.14)

We split the proof of the Proposition into two lemmas: The first lemma allows to compare almost deterministically with (which by rotation invariance of is distributed like ). denotes the image of the Poisson Point Process by , i.e. (recall (2.1))

(3.15)
Lemma 3.2.

Set , such that and for some , and that satisfies . Then for all sufficiently large one has

(3.16)

The second lemma estimates the probability that has the largest value among the different . The argument comes from [13],

Lemma 3.3.

For any value of and any

(3.17)

The proof for of the Proposition from the lemmas is straightforward with the use of Lemma A.1 that ensures that with probability the assumption on in Lemma 3.2 is satisfied.

Proof of Lemma 3.2.

By symmetry we can assume . The assumptions we have on guarantees that on the event , , and that . (see figure 1). Therefore using the strong Markov property for Brownian Motion,

(3.18)
Figure 1. Projection of the model along the two first coordinates. The two tubes represented are and . The shadowed region is , and this is where is modified. The full line that encircles denote the limit of , the region where we may have . The two trajectories represent typical trajectories of and . One can see on the picture that if is chosen large enough ( where is a constant depending on ), one the event , the hitting time of is larger than the hitting time of . Moreover the tube and the set are disjoint. Standard trigonometry allows to say that the maximal distance between a point in and is .

On the event , one has . Therefore, the right-hand side of (3.18) is smaller than

(3.19)

The first term on the above product is equal to . By the assumption one has on ( in the ball of radius ) , the second term is larger than

(3.20)

Hence the Lemma is proved if one can show that for all in

(3.21)

From our assumptions on , for large enough one has

(3.22)

and

(3.23)

(these inequality comes from the assumption one has taken for and trigonometry).

If one consider a -dimensional cube whose edges are parallel to the coordinate axis centered at and of side-length , then with probability , the exit time of the cube for a Brownian Motion started from is equal to . Moreover, if is large enough then this cube does not intersect (cf. (3.22) and (3.23)) and lies within the ball of radius . Hence (using symmetries of the cube)

(3.24)

where

(3.25)

and is the norm on . The hitting time is stochastically dominated by the first hitting time of by a one dimensional Brownian motion. And one has

(3.26)

Hence one has for large enough

(3.27)

Proof of Lemma 3.3.

Note that , are identically distributed variables. However they are not exchangeable, and therefore the statement is not that obvious. As

(3.28)

there exists some such that

(3.29)

Hence by rotational invariance of and

(3.30)

3.3. Change of environment: Adding traps in

With we construct a second environment that has more traps with radius in the region . The aim of this section is to show that typical event for are also typical for .

We construct and on the same probability space and for convenience denote by their joint probability. Recall that is Poisson Point Process in with intensity . Then define to be a Poisson Point Process on independent of with intensity

(3.31)

where denotes the Lebesgue measure on the set

(3.32)

and define

(3.33)

which is a Poisson Point Process on with intensity

(3.34)
Lemma 3.4.

Assume that

(3.35)

Then, there exists a constant not depending on such that for any event one has

(3.36)

Before going to the proof, we explain why the result holds: For the number of points in is a Poisson variable of mean

(3.37)

The fluctuation around the mean are therefore of order . The number of points in process is a Poisson variable of mean (intensity volume). Therefore the number of points one add to to get is of the same order as the fluctuation for the number of point of in , and for that reason the two process should typically look the same.

The result would not hold if had an intensity of a larger order.

Proof.

Let resp. denote the law of resp. under . For a function , we denote by resp. expectation w.r.t resp. . Note that is absolutely continuous with respect to and one has

(3.38)

For any event by Cauchy-Schwartz inequality, on has

(3.39)

What is left to show is that the first term in the right-hand side remains bounded with . One has

(3.40)

And

(3.41)

Note that the quantity is small uniformly in the domain of integration (by the assumption ) so that

(3.42)

Putting everything together one gets

(3.43)

3.4. The effect of the change of measure

In this section we estimate the difference between and .

Proposition 3.5.

Suppose that

(3.44)

Then, for any , with probability tending to one when goes to infinity

(3.45)

The idea of the proof is quite simple (see figure 2). Making the change of environment , we add roughly traps of radius larger than . The traps we add are wide enough so that every trajectory in has to go through every one of them (this explains our choice of adding only traps of large radius).

Figure 2. The shadowed region is , and this is where is modified. The full line that encircles denote the limit of , the region where the values taken by and may differ. A typical trajectory of is represented. The four dots denote , , the center of the traps that have been added. The zone of influence of these traps are represented as circles. The definition of implies that the radius of the traps are large enough to cover all the width of the tube on a segment of length .

Under , trajectories are roughly ballistic, so that they should typically spend a time of order in each trap. As the traps are of radius , they modify the potential by . Therefore, for most trajectories in , one should have

(3.46)

which heuristically explains the result.

To make this sketch rigorous, the main point is to give a proof of the fact that each trajectory spend a time of order in each trap. This is the aim of Proposition 3.6.

For a given function define the probability measure by

(3.47)

where is defined in the same way as with replaced by .

Given in one wants to check that most trajectories of spend a reasonable amount of time in the slice of the tube

(3.48)

Let denote the first coordinate of .

Proposition 3.6.

For any non-negative function such that for all such that , for any for large enough, and for any ,