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

Electronic Communications in Probability

Pablo Groisman¹ 

关键词: quasi-stationary distributions;    traveling waves;    branching random walk;    branching Lévy proceses;   

DOI  :  10.1214/18-ECP199

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 摘 要 】

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.

【 授权许可】

CC BY