Residual Diffusivity for Noisy Bernoulli Maps

TL;DR

This paper proves that certain chaotic maps derived from Bernoulli maps exhibit residual diffusivity, meaning their effective diffusivity remains positive even as the noise level approaches zero.

Contribution

It establishes residual diffusivity for a class of maps constructed from piecewise affine expanding Bernoulli maps, advancing understanding of noise effects in chaotic systems.

Findings

Residual diffusivity occurs for maps derived from Bernoulli maps.

Effective diffusivity remains positive as noise vanishes.

Provides mathematical proof for residual diffusivity in specified class of maps.

Abstract

Consider a discrete time Markov process $X^\varepsilon$ on $\mathbb R^d$ that makes a deterministic jump prescribed by a map $\varphi \colon \mathbb R^d \to \mathbb R^d$, and then takes a small Gaussian step of variance $\varepsilon^2$. For certain chaotic maps $\varphi$, the effective diffusivity of $X^\varepsilon$ may be bounded away from $0$ as $\varepsilon \to 0$. This is known as residual diffusivity, and in this paper we prove residual diffusivity occurs for a class of maps $\varphi$ obtained from piecewise affine expanding Bernoulli maps.