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

TL;DR

This paper proves the stability of traveling wave solutions in integro-differential equations linked to branching Markov processes, demonstrating the limiting behavior of the left-most particle through approximation by branching random walks.

Contribution

It establishes the stability of traveling waves for a class of integro-differential equations related to branching Markov processes, using approximation techniques.

Findings

Proves stability of traveling wave solutions.

Demonstrates limiting law of the left-most particle.

Uses approximation by branching random walk.

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 L\'{e}vy 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 A\"{i}d\'{e}kon [1] on the limiting law of the latter one.