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Front propagation and quasi-stationary distributions for one-dimensional Lévy processes

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, we show that the existence of a 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 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.