Front propagation and quasi-stationary distributions for one-dimensional L\'evy processes

TL;DR

This paper explores the connection between traveling waves in the F-KPP equation and quasi-stationary distributions of one-dimensional Le9vy processes, establishing their equivalence under certain conditions using probabilistic methods.

Contribution

It demonstrates the equivalence between the existence of traveling waves in the F-KPP equation and quasi-stationary distributions for Le9vy processes, extending known conditions in both areas.

Findings

Traveling wave existence is equivalent to quasi-stationary distribution existence.

Conditions for traveling waves and quasi-stationary distributions are extended.

Probabilistic methods link the two phenomena.

Abstract

We jointly investigate the existence of quasi-stationary distributions for one dimensional L\'evy processes and the existence of traveling waves for the Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation associated with the same motion. Using probabilistic ideas developed by S. Harris, we show that the existence of a traveling wave for the F-KPP equation associated with a centered L\'evy processes that branches at rate $r$ and travels at velocity $c$ is equivalent to the existence of a quasi-stationary distribution for a L\'evy process with the same movement but drifted by $-c$ and killed at zero, with mean absorption time $1/r$. This also extends the known existence conditions in both contexts. As it is discussed in a companion article, this is not just a coincidence but the consequence of a relation between these two phenomena.