Solutions of kinetic-type equations with perturbed collisions

Dariusz Buraczewski, Piotr Dyszewski and Alexander Marynych

Stochastic Processes and their Applications, 2023, vol. 159, issue C, 199-224

Abstract: We study a class of kinetic-type differential equations ∂ϕt/∂t+ϕt=Q̂ϕt, where Q̂ is an inhomogeneous smoothing transform and, for every t≥0, ϕt is the Fourier–Stieltjes transform of a probability measure. We show that under mild assumptions on Q̂ 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 Q̂. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as t→∞.

Keywords: Additive martingale; Branching random walk; Inhomogeneous smoothing transform; Kac model; Kinetic equation; Random trees (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spa.2023.01.014

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