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.