Self-similar solutions of kinetic-type equations: the boundary case

TL;DR

This paper investigates the long-term behavior of probability measures evolving under a kinetic-type equation with a smoothing transform, identifying a non-degenerate self-similar limit in the critical regime when standard normalization fails.

Contribution

It introduces a new scaling approach for the critical regime of kinetic equations, extending previous probabilistic representations to describe asymptotic self-similar solutions.

Findings

Identified the appropriate scaling for non-degenerate limits in the critical regime

Provided a probabilistic representation that refines previous methods

Described asymptotic properties of measures in the domain of attraction of stable laws

Abstract

For a time dependent family of probability measures $(\rho_t)_{t\ge 0}$ we consider a kinetic-type evolution equation $\partial \phi_t/\partial t + \phi_t = \widehat{Q} \phi_t$ where $\widehat{Q}$ is a smoothing transform and $\phi_t$ is the Fourier--Stieltjes transform of $\rho_t$. Assuming that the initial measure $\rho_0$ belongs to the domain of attraction of a stable law, we describe asymptotic properties of $\rho_t$, as $t\to\infty$. We consider the critical regime when the standard normalization leads to a degenerate limit and find an appropriate scaling ensuring a non-degenerate self-similar limit. Our approach is based on a probabilistic representation of probability measures $(\rho_t)_{t\ge 0}$ that refines the corresponding construction proposed in Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928--1961, 2012].