Solutions to kinetic-type evolution equations: beyond the boundary case

Dariusz Buraczewski, Konrad Kolesko, Matthias Meiners

Published 2019-09-01Version 1

We study the asymptotic behavior as $t \to \infty$ of a time-dependent family $(\mu_t)_{t \geq 0}$ of probability measures on ${\mathbb R}$ solving the kinetic-type evolution equation $\partial_t \mu_t + \mu_t = Q(\mu_t)$ where $Q$ is a smoothing transformation on ${\mathbb R}$. This problem has been investigated earlier, e.g.\ by Bassetti and Ladelli [\emph{Ann. Appl. Probab.} 22(5): 1928--1961, 2012] and Bogus, Buraczewski and Marynych [To appear in \emph{Stochastic Process. Appl.}]. Combining the refined analysis of the latter paper providing a probabilistic description of the solution $\mu_t$ as the law of a suitable random sum related to a continuous-time branching random walk at time $t$ with recent advances in the analysis of the extremal positions in the branching random walk we are able to solve the remaining case that has not been addressed before.