Brownian flow on a finite interval with jump boundary conditions

Abstract

We consider a stochastic flow in an interval $[-a,b]$, where $a,b>0$. Each point of the interval is driven by the same Brownian path and jumps to zero when it reaches the boundary of the interval. Assuming that $a/b$ is irrational we study the long term behavior of a random measure $\mu_t$, the image of a finite Borel measure $\mu_0$ under the flow. We show that if $\mu_0$ is absolutely continuous with respect to the Lebesgue measure then the time averages of the variance of $\mu_t$ converge to zero almost surely. We also prove that for an arbitrary finite Borel measure $\mu_0$ the Lebesgue measure of the support of $\mu_t$ decreases to zero as $t\to\infty$ with probability one.