Open Access
2018 Front propagation and quasi-stationary distributions for one-dimensional Lévy processes
Pablo Groisman, Matthieu Jonckheere
Electron. Commun. Probab. 23: 1-11 (2018). DOI: 10.1214/18-ECP199


We jointly investigate the existence of quasi-stationary distributions for one dimensional Lévy 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 [16], we show that the existence of a monotone traveling wave for the F-KPP equation associated with a centered Lévy processes that branches at rate $r$ and travels at velocity $c$ is equivalent to the existence of a quasi-stationary distribution for a Lévy process with the same movement but drifted by $-c$ and killed at the first entry into the negative semi-axis, with mean absorption time $1/r$. This also extends the known existence conditions in both contexts. As it is discussed in [15], this is not just a coincidence but the consequence of a relation between these two phenomena.


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Pablo Groisman. Matthieu Jonckheere. "Front propagation and quasi-stationary distributions for one-dimensional Lévy processes." Electron. Commun. Probab. 23 1 - 11, 2018.


Received: 21 November 2016; Accepted: 21 November 2018; Published: 2018
First available in Project Euclid: 15 December 2018

zbMATH: 07023479
MathSciNet: MR3896831
Digital Object Identifier: 10.1214/18-ECP199

Primary: 60G51 , 60J68 , 60J80

Keywords: branching Lévy proceses , Branching random walk , Quasi-stationary distributions , Traveling waves

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