[Paper Review] Limit theorems for random walks in dynamic random environment

Frank Redig, Florian Völlering|arXiv (Cornell University)|Jun 21, 2011

Stochastic processes and statistical mechanics9 references8 citations

TL;DR

This paper establishes limit theorems for random walks in dynamic random environments by introducing a coupling-based condition on the environment's mixing properties. Under this condition, the environment process inherits strong ergodicity and concentration properties, enabling rigorous proofs of the law of large numbers, central limit theorem, Einstein relation, and recurrence/transience criteria for the walker's position, with continuous dependence on jump rates.

ABSTRACT

We study a general class of random walks driven by a uniquely ergodic Markovian environment. Under a coupling condition on the environment we obtain strong ergodicity properties and concentration inequalities for the environment as seen from the position of the walker, i.e the environment process. We also obtain ergodicity of the uniquely ergodic measure of the environment process as well as continuity as a function of the jump rates of the walker. As a consequence we obtain several limit theorems, such as law of large numbers, Einstein relation, central limit theorem and concentration properties for the position of the walker.

Motivation & Objective

• To establish limit theorems for random walks in dynamic random environments when the environment is uniquely ergodic and mixes sufficiently fast.
• To transfer ergodic and mixing properties from the environment to the environment process seen from the walker’s position.
• To develop a new formalism based on backwards martingales to derive concentration inequalities for the walker’s position.
• To prove continuity of the asymptotic speed and covariance matrix with respect to jump rates.
• To derive recurrence and transience criteria in one dimension based on the sign of the asymptotic speed.

Proposed method

• Introduces a coupling condition on the environment such that the distance between initial configurations decays fast enough that $ t^d $ times the decay is integrable in time.
• Uses the coupling condition to prove that the environment process also admits a coupling with integrable decay, losing at most a factor of $ t^d $.
• Applies a novel formalism of non-time-homogeneous backwards martingales to derive exponential moment bounds and concentration inequalities for additive functionals of the environment process.
• Establishes the environment process as a Markov process and uses its ergodicity to decompose the walker’s position into a martingale and a controlled additive functional.
• Employs the martingale approach in the Appendix to prove a functional central limit theorem and variance control for the walker’s position.
• Uses concentration inequalities to derive recurrence and transience results in $ d=1 $, depending on whether the asymptotic speed is zero or non-zero.

Experimental results

Research questions

• RQ1 Under what conditions on the dynamic environment does the environment process inherit strong ergodicity and mixing properties?
• RQ2 How can concentration inequalities for the walker’s position be rigorously derived when the position is not a standard additive functional of the environment process?
• RQ3 What conditions ensure the law of large numbers and central limit theorem for the random walk in a dynamic environment?
• RQ4 How does the asymptotic speed of the walker depend continuously on the jump rates of the walk?
• RQ5 What determines recurrence or transience of the walk in one dimension, and how can this be linked to the environment’s mixing and ergodic properties?

Key findings

• The environment process inherits ergodicity and mixing properties from the environment under a coupling condition, with decay rates losing at most a factor of $ t^d $.
• A law of large numbers holds for the walker’s position with an asymptotic speed that depends continuously on the jump rates of the walk.
• A functional central limit theorem is established, with a controllable asymptotic covariance matrix derived from the environment process’s variance.
• An Einstein relation is proven, linking the derivative of the speed with respect to perturbations in the environment to the diffusion matrix.
• Concentration inequalities for the walker’s position are derived using backwards martingales, enabling rigorous recurrence and transience results in $ d=1 $.
• In one dimension, the walk is recurrent if the asymptotic speed is zero and transient if the speed is non-zero, under the given coupling and ergodicity assumptions.