1. Solutions of kinetic-type equations with perturbed collisions
- Author
-
Buraczewski, Dariusz, Dyszewski, Piotr, and Marynych, Alexander
- Subjects
Statistics and Probability ,Applied Mathematics ,Modeling and Simulation ,Probability (math.PR) ,FOS: Mathematics ,Primary: 60J85, Secondary: 82C40, 60F05 ,Mathematics - Probability - 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 $t\to\infty$., Comment: 24 pages, to appear in Stochastic Processes and Their Applications
- Published
- 2022
- Full Text
- View/download PDF