On stability of traveling wave solutions for integro-differential equations related to branching Markov processes

Abstract

The aim of this paper is to prove stability of traveling waves for integro-differential equations connected with branching Markov processes. In other words, the limiting law of the left-most particle of a (time-continuous) branching Markov process with a Lévy non-branching part is demonstrated. The key idea is to approximate the branching Markov process by a branching random walk and apply the result of Aïdékon [Ann. Probab. 41 (2013) 1362–1426] on the limiting law of the latter one.