Invariance principle for diffusions in random environment

In this article we are going to show an invariance principle, i.e. convergence of a process to Brownian motion under a space-time scaling, for the diffusive motion of a particle interacting with infinitely many other particles in R, d > 2. To this end we will use a general approach developed by A. De Masi, P.A. Ferrari, S. Goldstein, and W.D. Wick [3] and C. Kipnis and S.R.S. Varadhan [5]. The random environment we consider is a random configuration γ, i.e. a locally finite subset of R, chosen with respect to a grand canonical Gibbs measure μE with respect to a potential VE. A particle x performs a diffusive motion (Xt)t>0 in R d \ γ and interacts with the (frozen) points in γ via a symmetric, translation invariant pair potential VI. De Masi et al. also discussed this situation in the case of a positive, compactly-supported C interaction potential. Furthermore, they assumed VI = VE. We will assume that VI has a singularity at the origin, that is repulsion of particles at distance zero, and a proper decay at infinity. But it has to be neither positive nor of finite range. The case that VI is the Lennard-Jones potential is included. Furthermore, it does not have to be the same potential VE to which the Gibbs measure corresponds. By using newer results (cf. [7,10]) on diffusions in random environment we can construct the corresponding stochastic process in this setting, which describes the motion of the particle in the random environment, and then apply the technique by De Masi et al. to obtain an invariance principle. The main idea of this approach is to write (Xt)t>0 as a functional of its environment process (ξt)t>0, i.e. the motion of the environment as seen from the particle. (ξt)t>0 is a process on the configuration space Γ. De Masi et al. formulated conditions on this environment process which imply an invariance principle for the original process (Xt)t>0.


Introduction
In this article we are going to show an invariance principle, i.e. convergence of a process to Brownian motion under a space-time scaling, for the diffusive motion of a particle interacting with infinitely many other particles in R d , d 2. To this end we will use a general approach developed by A. De Masi, P.A. Ferrari, S. Goldstein, and W.D. Wick [3] and C. Kipnis and S.R.S. Varadhan [5].
The random environment we consider is a random configuration γ, i.e. a locally finite subset of R d , chosen with respect to a grand canonical Gibbs measure µ E with respect to a potential V E .A particle x performs a diffusive motion (X t ) t 0 in R d \ γ and interacts with the (frozen) points in γ via a symmetric, translation invariant pair potential V I .De Masi et al. also discussed this situation in the case of a positive, compactly-supported C ∞ interaction potential.Furthermore, they assumed V I = V E .We will assume that V I has a singularity at the origin, that is repulsion of particles at distance zero, and a proper decay at infinity.But it has to be neither positive nor of finite range.The case that V I is the Lennard-Jones potential is included.Furthermore, it does not have to be the same potential V E to which the Gibbs measure corresponds.By using newer results (cf.[7,10]) on diffusions in random environment we can construct the corresponding stochastic process in this setting, which describes the motion of the particle in the random environment, and then apply the technique by De Masi et al. to obtain an invariance principle.The main idea of this approach is to write (X t ) t 0 as a functional of its environment process (ξ t ) t 0 , i.e. the motion of the environment as seen from the particle.(ξ t ) t 0 is a process on the configuration space Γ.De Masi et al. formulated conditions on this environment process which imply an invariance principle for the original process (X t ) t 0 .

Configuration spaces
Here we will briefly recall the framework of configuration space analysis.For a general overview we refer to e.g.[1,2].
The configuration space Γ := Γ(R d ) is defined as the set of all locally finite subsets of R d , i.e.
Here |γ ∩ K| denotes the cardinality of the set γ ∩ K.The sets γ ∈ Γ are called configurations.We will think of a configuration γ as a set of indistinguishable particles in R d .Every γ ∈ Γ can be identified with a positive Radon measure on R d via where δ y denotes the Dirac measure with mass in y ∈ R d , and the empty configuration is identified with the zero measure.Then, as a subset of the set of all positive Radon measures, Γ is naturally topologized by the (subspace topology induced by the) vague topology, i.e., the weakest topology on Γ such that all maps are continuous, where f ∈ C 0 (R d ), i.e., f is a continuous function on R d with compact support.Let B(Γ) denote the Borel σ-algebra on Γ corresponding to this topology.In particular Consider a pair potential V , i.e., a Lebesgue-measurable function The relative energy of interaction between a particle at point x ∈ R d and a configuration γ ∈ Γ via the potential V is defined as A probability measure µ on (Γ, B(Γ)) is called a (grand canonical) Gibbs measure with respect to activity parameter z > 0 and potential V if it satisfies the Georgii-Nguyen-Zessin identity (GNZ): for any positive There exist equivalent definitions of Gibbs measures, e.g.via Dobrushin-Lanford-Ruelle approach.
A system of non-negative measurable symmetric functions k A constant ξ > 0 is called Ruelle bound if it satisfies It is well-known that tempered Gibbs measures corresponding to superstable, lower regular, integrable (in the sense of ( 10)) pair potentials have correlation functions with Ruelle bound, cf.[11].

Construction of the process
We want to describe the diffusive motion of a particle x ∈ R \ γ, where γ ∈ Γ is a configuration randomly chosen with respect to some Gibbs measure.From now on assume that d 2, then R d \γ is connected for any γ.
Consider a symmetric, translation invariant pair potential V , i.e., V (x − y) = Ṽ (|x − y|), x, y ∈ R d , for some proper function Ṽ : (0, +∞) → R. We assume repulsion at zero distance and decay of the potential at infinity, i.e., Assume that the relative energy E V (x, γ) is finite for every x ∈ R d \ γ. (This holds under the assumptions of Theorem 1 and Theorem 2 below.)Let the logarithmic derivative of ρ γ , is well-defined Lebesgue-a.e.The motion of the particle in R d \ γ should be described by the following stochastic differential equation (SDE): Here W t is a standard Brownian motion on R d .Define the set of admissible configurations.Here B(x, r) denotes the ball with center x and radius r.For many classes of probability measures ν on Γ we have ν(Γ ad ) = 1, in particular for Ruelle measures corresponding to superstable pair potentials, cf.[6,9].
The following result is due to N.V. Krylov and M. Röckner [10].
Theorem 1. Assume that the potential V satisfies (4) and the following assumptions: in the sense of distributions on R d \ {0}, where U (x) := c(1 Let γ ∈ Γ ad .Then (5) has a unique strong solution defined for all times if Yu.G.Kondratiev, A.Yu. Konstantinov and M. Röckner [7] have discussed earlier the slightly different stochastic differential equation with positive definite diffusion operator (H, C ∞ 0 (R d )), Hu = −∆u − β, ∇u , on L 2 (R d , ρ γ dx).They proved the following result: Theorem 2. Assume that V satisfies the following conditions: and V −a for some a > 0; Furthermore, they constructed a weak solution for (9) under the conditions of Theorem 2.
Lemma 3. Assume that V satisfies the conditions from Theorem 2. Then ∇V ∈ L 1 (R d , exp(−V )dx).Furthermore, V satisfies the integrability condition for every (inverse temperature) p 1.
Proof.We have that With condition (ii) from Theorem 2 this implies the first assertion.
For the second part let p 1. Of course, Outside B(0, k 0 ) we have Hence, the assertion follows.
Assume that V satisfies the conditions from Theorem 2. Then the operator H is L 2 -unique or, equivalently, essentially self-adjoint (cf., e.g., [4,Corollary 1.2]).Hence, the corresponding pre-Dirichlet form on L 2 (R d , ρ γ dx) has a unique Dirichlet extension (E, D(E)).Since we assume that the dimension d 2, we have that is connected.Therefore, we can apply [4, Theorem 3.7 (i)] plus a localization argument to obtain the ergodicity of the corresponding semigroup.

Environment process and invariance principle
For x 0 ∈ R d define the space shift by x 0 on Γ via Then, for a process X t ∈ R d \ γ, t 0, the corresponding environment process is defined as ξ γ is a process on Γ starting in γ − X 0 .
Let V E (E stands for environment) be a symmetric, translation invariant potential on R d , and let µ E be an extremal grand canonical Gibbs measure on (Γ, B(Γ)) with respect to V E and activity parameter z > 0, which is invariant and ergodic with respect to space shifts Θ x0 .We assume, that µ E has correlation functions k (n) µE with Ruelle bound ξ > 0. Furthermore, let V I satisfy the assumptions of Theorem 1 and Theorem 2.
The space shift Θ x0 acts on the relative energy via Because of the integrability condition (10) for any inverse temperature, in particular for p, we have that n ∈ N. From the Ruelle bound it follows This proves the first part of the assertion.
However, the term C := e ξ θx 0 L 1 in the last equation is independent of x 0 , since, by translation invariance of the Lebesgue measure, θ x0 L 1 = θ 0 L 1 for all x 0 .Thus, sup where we obtain that µ E and µ * are equivalent measures, i.e., they have the same nullsets.