Publication 25-CNA-012

Residual Diffusivity For Expanding Bernoulli Maps

William Cooperman
Department of Mathematics
ETH Zürich, Switzerland
bill@cprmn.org

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu

James Nolen
Department of Mathematics
Duke University
Durham, NC 27708
james.nolen@duke.edu

Abstract: Consider a discrete time Markov process $X^\epsilon$ on $\mathbb{R}^d$ 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.

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