Residual Diffusivity for Expanding Bernoulli Maps

TL;DR

This paper investigates residual diffusivity in a Markov process with deterministic jumps and Gaussian noise, showing that certain chaotic jumps lead to a non-vanishing asymptotic variance as noise diminishes.

Contribution

It proves that residual diffusivity occurs in processes with expanding Bernoulli map jumps, revealing a new class of systems with persistent variance at small noise levels.

Findings

Residual diffusivity occurs in systems with chaotic jumps.

Expanding Bernoulli maps induce bounded asymptotic variance.

Asymptotic variance remains positive as noise approaches zero.

Abstract

Consider a discrete time Markov process $X^\epsilon$ on $\mathbf R^d$ that makes a deterministic jump based on its current location, and then takes a small Gaussian step of variance $\epsilon^2$. We study the behavior of the asymptotic variance as $\epsilon \to 0$. In some situations (for instance if there were no jumps), then the asymptotic variance vanishes as $\epsilon \to 0$. When the jumps are "chaotic", however, the asymptotic variance may be bounded from above and bounded away from $0$, as $\epsilon \to 0$. This phenomenon is known as residual diffusivity, and we prove this occurs when the jumps are determined by certain expanding Bernoulli maps.