Using Bernoulli maps to accelerate mixing of a random walk on the torus

We study the mixing time of a random walk on the torus, alternated with a Lebesgue measure preserving Bernoulli map. Without the Bernoulli map, the mixing time of the random walk alone is $O(1\mathrm{/}{\epsilon }^2)$, where $\epsilon$ is the step size. Our main results show that for a class of Bernoulli maps, when the random walk is alternated with the Bernoulli map $\phi$ the mixing time becomes $O(\mathrm{\mid }\ln \epsilon \mathrm{\mid })$. We also study the \emph{dissipation time} of this process, and obtain $O(\mathrm{\mid }\ln \epsilon \mathrm{\mid })$ upper and lower bounds with explicit constants.