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Central limit theorem for random walks in divergence free random drift field -- revisited

Bálint Tóth

TL;DR

This work provides a partly alternative, functional-analytic proof of the central limit theorem for random walks in divergence-free random drift fields, under the $\mathcal{H}_{-1}$ condition and ellipticity of the symmetric part. It replaces the diagonal heat-kernel bound (derived from Nash’s inequality) with a direct Kipnis-Varadhan-type martingale-approximation argument tailored to a non-self-adjoint generator, and it relies on the structure of the Riesz transforms and a crucial essential skew-self-adjointness of the operator $|\Delta|^{-1/2}\nabla^* H \nabla |\Delta|^{-1/2}$. The core technical contribution is verifying the skew-self-adjointness via truncation and weak limits, which yields the required martingale approximation and hence the CLT, with a subdiffusive error term for the remainder. The approach is explicitly designed to be extendable to non-elliptic settings, forming a foundation for forthcoming results beyond the elliptic regime.

Abstract

In [Kozma-Toth, Ann. Probab. v 45, pp 4307-4347 (2017)] the weak CLT was established for random walks in doubly stochastic (or, divergence-free) random environments, under the following conditions: 1. Strict ellipticity assumed for the symmetric part of the drift field. 2. $H_{-1}$ assumed for the antisymmetric part of the drift field. The proof relied on a martingale approximation (a la Kipnis-Varadhan) adapted to the non-self-adjoint and non-sectorial nature of the problem. The two substantial technical components of the proof were: 1. A functional analytic statement about the unbounded operator formally written as $|L+L^*|^{-1/2}(L-L^*)|L+L^*|^{-1/2}$, where $L$ is the infinitesimal generator of the environment process, as seen from the position of the moving random walker. 2. A diagonal heat kernel upper bound which follows directly from Nash's inequality, or, alternatively, from the "evolving sets" arguments of [Morris-Peres, Probab. Theory Rel. Fields. v. 133 pp 245-266 (2005)] valid only under the assumed strict ellipticity. In this note we present a partly alternative proof of the same result which relies only on functional analytic arguments and not on the diagonal heat kernel upper bound provided by Nash's inequality. This alternative proof is relevant since it can be naturally extended to non-elliptic settings pushed to the optimum, which will be presented in a forthcoming paper. The goal of this note is to present the argument in its simplest and most transparent form.

Central limit theorem for random walks in divergence free random drift field -- revisited

TL;DR

This work provides a partly alternative, functional-analytic proof of the central limit theorem for random walks in divergence-free random drift fields, under the condition and ellipticity of the symmetric part. It replaces the diagonal heat-kernel bound (derived from Nash’s inequality) with a direct Kipnis-Varadhan-type martingale-approximation argument tailored to a non-self-adjoint generator, and it relies on the structure of the Riesz transforms and a crucial essential skew-self-adjointness of the operator . The core technical contribution is verifying the skew-self-adjointness via truncation and weak limits, which yields the required martingale approximation and hence the CLT, with a subdiffusive error term for the remainder. The approach is explicitly designed to be extendable to non-elliptic settings, forming a foundation for forthcoming results beyond the elliptic regime.

Abstract

In [Kozma-Toth, Ann. Probab. v 45, pp 4307-4347 (2017)] the weak CLT was established for random walks in doubly stochastic (or, divergence-free) random environments, under the following conditions: 1. Strict ellipticity assumed for the symmetric part of the drift field. 2. assumed for the antisymmetric part of the drift field. The proof relied on a martingale approximation (a la Kipnis-Varadhan) adapted to the non-self-adjoint and non-sectorial nature of the problem. The two substantial technical components of the proof were: 1. A functional analytic statement about the unbounded operator formally written as , where is the infinitesimal generator of the environment process, as seen from the position of the moving random walker. 2. A diagonal heat kernel upper bound which follows directly from Nash's inequality, or, alternatively, from the "evolving sets" arguments of [Morris-Peres, Probab. Theory Rel. Fields. v. 133 pp 245-266 (2005)] valid only under the assumed strict ellipticity. In this note we present a partly alternative proof of the same result which relies only on functional analytic arguments and not on the diagonal heat kernel upper bound provided by Nash's inequality. This alternative proof is relevant since it can be naturally extended to non-elliptic settings pushed to the optimum, which will be presented in a forthcoming paper. The goal of this note is to present the argument in its simplest and most transparent form.
Paper Structure (12 sections, 6 theorems, 70 equations)

This paper contains 12 sections, 6 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $\mathcal{H}_{-1}$-condition
  4. The CLT
  5. Comments, history, examples
  6. Spaces and operators
  7. The infinitesimal generator
  8. Basic spaces and operators
  9. The space $\mathcal{H}_-$ and the Riesz operators
  10. Proof of Theorem \ref{['thm: main']}
  11. Kipnis-Varadhan theory - the abstract form
  12. Proof of Theorem \ref{['thm: main']}

Key Result

Theorem 1.1

(kozma-toth-17 Theorem 1) Assume h-tensor, h-l2, and v-is-curl-of-h. Then the process $t\mapsto I(t)\in\mathbb R^d$ on the right hand side of martingale decomposition is decomposed as so that $t\mapsto Z(t)\in\mathbb R^d$ is a quenched martingale whose increments are stationary, ergodic and square integrable in the annealed setting, and The martingales $t\mapsto Y(t)$ and $t\mapsto Z(t)$ do not

Theorems & Definitions (8)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 3.1
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof