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Brownian flow on a finite interval with jump boundary conditions
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.