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

TL;DR

This paper analyzes the long-term behavior of solutions to a kinetic-type evolution equation involving a smoothing transformation, extending previous results by addressing previously unresolved cases through probabilistic and branching random walk techniques.

Contribution

It provides a comprehensive analysis of the asymptotic behavior of solutions beyond the boundary case, weakening existing assumptions for existence and uniqueness.

Findings

Resolved the remaining open case in the asymptotic analysis.

Connected the solution behavior to extremal positions in branching random walks.

Weakened conditions needed for solution existence and uniqueness.

Abstract

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 [Ann. Appl. Probab. 22(5): 1928-1961, 2012] and Bogus, Buraczewski and Marynych [Stochastic Process. Appl. 130(2):677-693, 2020]. Combining the refined analysis of the latter paper, which provides 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 been left open until now. In the course of our work, we significantly weaken the assumptions in the literature that guarantee the existence (and uniqueness) of a solution to the evolution equation $\partial_t \mu_t + \mu_t = Q(\mu_t)$.