Theoretical Statistics and Mathematics Unit, ISI Delhi

Long term behavior of a Brownian flow with jumps

by Siva Athreya, Elena Kosygina and Steve Tanner

We consider a stochastic jump flow in an interval (−a, b), where $a, b > 0$. Each particle
of the flow performs a canonical Brownian motion and jumps to zero when it reaches −a or
b. We study the long term behavior of a random measure $\mu_t$ which is the image of a finite
Borel measure $\mu_0$ under the flow. When $a/b$ is irrational, we show that for almost every
driving Brownian path the time averages of the variance of $\mu_t$ converge to zero, and the
Lebesgue measure of the support of $\mu_t$ decreases to zero as time goes to infinity. When $a/b$ is
rational, we show that the Lebesgue measure of the support of $\mu_t$ decreases to its minimum
value in finite time almost surely. In addition, if $\mu_0$ is proportional to Lebesgue measure we
show that the number of connected components of the support of $\mu_t$ is a recurrent process,
which assumes every positive integer value with probability 1.

isid/ms/2002/01 [fulltext]

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