Solutions of kinetic-type equations with perturbed collisions

TL;DR

This paper investigates solutions to kinetic-type equations involving inhomogeneous smoothing transforms, establishing existence, uniqueness, and asymptotic behavior of solutions through stochastic process representations.

Contribution

It introduces a novel approach to solving kinetic equations with perturbed collisions by linking solutions to stochastic processes and limit theorems.

Findings

Unique solutions exist under mild conditions.

Solutions are represented as characteristic functions of stochastic processes.

Asymptotic properties are characterized as time approaches infinity.

Abstract

We study a class of kinetic-type differential equations $\partial \phi_t/\partial t+\phi_t=\widehat{\mathcal{Q}}\phi_t$, where $\widehat{\mathcal{Q}}$ is an inhomogeneous smoothing transform and, for every $t\geq 0$, $\phi_t$ is the Fourier--Stieltjes transform of a probability measure. We show that under mild assumptions on $\widehat{\mathcal{Q}}$ the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk pertaining to $\widehat{\mathcal{Q}}$. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as $t\to\infty$.