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

Xi Geng, Mihai Gradinaru, Samy Tindel

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.

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

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.

Paper Structure

This paper contains 46 sections, 62 theorems, 850 equations.

Table of Contents

  1. Introduction
  2. Sinai's random walk
  3. Preliminaries on Sinai's walk
  4. Definition of Sinai's walk
  5. Some martingale problems
  6. Rescaled version of Sinai's walk
  7. Mild formulation of a discrete PDE
  8. Heuristic considerations and rescaling
  9. Estimates for the discrete heat kernel
  10. Brownian motion in Brownian environment
  11. Definition of the Brox diffusion
  12. Heuristics about the PDE problem related to Brox diffusion
  13. The rough path structure for the fixed point problem
  14. Metric on the Brownian rough path $W$
  15. The solution space and the corresponding rough path metric
  16. ...and 31 more sections

Key Result

Theorem 1.3

Let $X^\delta$ be the lazy version of Sinai's random walk as described in Section sec:prelim-sinai, properly rescaled as in Section rescaled-Sinai-walk. Consider the weak solution $X^c$ to the equation Broxdiff-int, as well as a function $h\in{\rm C}_{b}^{3}$. Then there exists a coupling $(X^\delta where $C_{h,T}(\omega)$ is a random constant which only depends on $h,T$ and the environment $\omeg

Theorems & Definitions (137)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Remark 2.7
  • ...and 127 more