Stationary Distribution of Random Motion with Delay in Reflecting Boundaries

Abstract

In this paper we study a continuous time random walk in the line with two boundaries [a,b], a < b. The particle can move in any of two directions with different velocities v 1 and v 2 . We consider a special type of boundary which can trap the particle for a random time. We found closed-form expressions for the stationary distribution of the position of the particle not only for the alternating Markov process but also for a broad class of semi-Markov processes.

Telegraph process in the bounded domain with absorbing lower boundary and reflecting with delay upper boundary

“…Unlike [16], we assume that reflection is not instantaneous -point stays in the upper boundary until switch to the regime 0 occurs. That type of boundary conditions was analyzed in [1] and the dynamics for the case of two boundaries of this type was described in [20]. However, the case of mixed boundaries, to the best our knowledge, was not yet analyzed in the literature.…”