Dynamic Random Environment Research Articles

A central limit theorem for Gibbs measures relative to Brownian motion

We study a Gibbs measure over Brownian motion with a pair potential which depends only on the increments. Assuming a particular form of this pair potential, we establish that in the infinite volume limit the Gibbs measure can be viewed as Brownian motion moving in a dynamic random environment. Thereby we are in a position to use the technique of Kipnis and Varadhan and to prove a functional central limit theorem.

Random walk in dynamic environment with mutual influence

We study a discrete time random walk in Z v in a dynamic random environment, when the evolution of the environment depends on the random walk (mutual influence). We assume that the unperturbed environment evolves independently at each site, as an ergodic Markov chain, and that the interaction is strictly local. We prove that the central limit theorem for the position X t of the random walk (particle) holds, whenever one of the following conditions is met: (i) the particle cancels the memory of the environment and the influence of the environment on the random walk is small; (ii) the exponential relaxation rate of the environment is large; (iii) the mutual interaction of the environment and the random walk is small. We also prove convergence of the distribution of the ‘environment as seen from the particle’. Proofs are obtained by cluster expansion techniques.