Residual Diffusivity for Expanding Bernoulli Maps

TL;DR

The paper proves residual diffusivity for a discrete-time Markov system X^ε_n that evolves by a deterministic jump φ and a small Gaussian perturbation εξ_{n+1}. By focusing on expanding Bernoulli maps φ with exponential mixing, the authors decompose the trajectory increments into independent and dependent parts (S,J,R) and use a coupling argument to show the long-time variance limit, as ε→0, matches a variance computed under a uniform initial distribution. A key step is proving that the Bernoulli structure makes floor-increments independent, allowing explicit calculation of the limiting variance in terms of ⌊X^0_1⌋ under the uniform measure on the unit cube. The results provide a rigorous, explicit instance of residual diffusivity in a chaotic, discretized dynamical system, linking to homogenization and effective diffusivity theory and offering concrete mixing-time bounds for the auxiliary toral process.

Abstract

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

Figures (2)

• Figure 1: One example of an expanding Bernoulli map $\varphi$. The colored regions on the left are mapped to regions of the same color on the right.
• Figure 2: Asymptotic variance for the doubling map, and the shifted doubling map as $\varepsilon$ varies from $10^{-1}$ to $10^{-5}$. For the doubling map the asymptotic variance approaches the value on the right of \ref{['e:residual-diffusivity']} (blue dashed line) as $\varepsilon \to 0$. The shifted doubling map, however, doesn't exhibit residual diffusivity and the asymptotic variance approaches $0$ instead of the value on the right of \ref{['e:residual-diffusivity']} (orange dashed line).

Theorems & Definitions (15)

• Theorem 1.1
• Remark 2.2
• Lemma 3.1
• Lemma 3.2
• proof : Proof of Theorem \ref{['t:residual-diffusivity']}
• proof : Proof of Lemma \ref{['l:subGaussianNorm']}
• Lemma 4.1
• Proposition 4.2
• proof : Proof of Lemma \ref{['l:UnifID']}
• proof : Proof of Lemma \ref{['l:varS']}