Solutions of kinetic-type equations with perturbed collisions

Abstract

We study a class of kinetic-type differential equations $\partial {\varphi }_t/\partial t+{\varphi }_t=\hat{\mathcal{Q}}{\varphi }_t$, where $\hat{\mathcal{Q}}$ is an inhomogeneous smoothing transform and, for every $t\ge 0$, ${\varphi }_t$ is the Fourier--Stieltjes transform of a probability measure. We show that under mild assumptions on $\hat{\mathcal{Q}}$ 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 $\hat{\mathcal{Q}}$. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as $t\to \infty$.

arXiv comment24 pages, published in Stochastic Processes and Their Applications

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