Markov Processes with Redistribution

#### I. Grigorescu, Min Kang

2013, v.19, Issue 3, 497-520

ABSTRACT

We study a class of stochastic processes evolving in the interior of a set $D$ according to an underlying Markov kernel, undergoing jumps to a random point $x$ in $D$ with distribution $\nu_{\xi}(dx)$ as soon as they reach a point $\xi$ outside $D$. We give conditions on the family of measures $\nu_{\xi}(dx)$ preventing that infinitely many jumps occur in finite time (explosiveness), conditions for ergodicity and the existence of a spectral gap. The setup is applied to a multitude of models considered recently, including particle systems like the Fleming - Viot branching process and a new variant of the Bak - Sneppen dynamics from evolutionary biology. The last part of the paper is expository and discusses the relation with quasi-stationary distributions.

Keywords: Doeblin condition,jump diffusion process,Fleming - Viot branching process,quasi-stationary distribution (qsd)