A coupling between random walks in random environments and Brox's diffusion

TL;DR

This paper establishes a specific coupling between Sinai's random walk and Brox's diffusion, providing a quantitative measure of their convergence using martingale problem techniques in the rough path framework.

Contribution

It introduces a novel coupling method that explicitly links Sinai's walk with Brox's diffusion, advancing the understanding of their convergence behavior.

Findings

Quantifies convergence between Sinai's walk and Brox's diffusion.

Uses martingale problem convergence in rough path setting.

Provides a new coupling construction for these stochastic processes.

Abstract

It is known that a properly rescaled version of Sinai's random walk converges in distribution to Brox's diffusion. In this article we quantify this convergence by considering a specific coupling between Sinai's walk and Brox's diffusion. Our method relies on convergence results for martingale problems considered in the rough path setting.