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July  2006, 6(4): 867-880. doi: 10.3934/dcdsb.2006.6.867

Brownian flow on a finite interval with jump boundary conditions

Elena Kosygina 1,

1. 

Department of Mathematics, Box B6-230, Baruch College - CUNY, One Bernard Baruch Way, New York, NY 10010, United States

Received  February 2005 Revised  November 2005 Published  April 2006

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.
Keywords: stationary ergodic measure, Hopf's ratio ergodic theorem., Brownian flow, embedded Markov chain.
Mathematics Subject Classification: Primary: 60J65, 60F15; Secondary: 60J1.
Citation: Elena Kosygina. Brownian flow on a finite interval with jump boundary conditions. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 867-880. doi: 10.3934/dcdsb.2006.6.867
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