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Solutions of kinetic-type equations with perturbed collisions

Author

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  • Buraczewski, Dariusz
  • Dyszewski, Piotr
  • Marynych, Alexander

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→∞.

Suggested Citation

  • Buraczewski, Dariusz & Dyszewski, Piotr & Marynych, Alexander, 2023. "Solutions of kinetic-type equations with perturbed collisions," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 199-224.
    • Handle: RePEc:eee:spapps:v:159:y:2023:i:c:p:199-224
    DOI: 10.1016/j.spa.2023.01.014
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    References listed on IDEAS

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    1. Bogus, Kamil & Buraczewski, Dariusz & Marynych, Alexander, 2020. "Self-similar solutions of kinetic-type equations: The boundary case," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 677-693.
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