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

Abstract

For a time dependent family of probability measures $({\rho }_t{)}_{t\ge 0}$ we consider a kinetic-type evolution equation $\partial {\varphi }_t/\partial t+{\varphi }_t=\hat{Q}{\varphi }_t$ where $\hat{Q}$ is a smoothing transform and ${\varphi }_t$ 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].

to appear in Stochastic Processes and their Applications