Residual Diffusivity for Noisy Bernoulli Maps

Consider a discrete time Markov process $X^{\epsilon }$ on ${\mathbb{R}}^d$ that makes a deterministic jump prescribed by a map $\phi \text{\hspace{0.17em}⁣}:{\mathbb{R}}^d\to {\mathbb{R}}^d$, and then takes a small Gaussian step of variance ${\epsilon }^2$. For certain chaotic maps $\phi$, the effective diffusivity of $X^{\epsilon }$ may be bounded away from $0$ as $\epsilon \to 0$. This is known as residual diffusivity, and in this paper we prove residual diffusivity occurs for a class of maps $\phi$ obtained from piecewise affine expanding Bernoulli maps.