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Self-similar solutions of kinetic-type equations: The boundary case

Author

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  • Bogus, Kamil
  • Buraczewski, Dariusz
  • Marynych, Alexander

Abstract

For a time dependent family of probability measures (ρt)t⩾0 we consider a kinetic-type evolution equation ∂ϕt∕∂t+ϕt=Q̂ϕt where Q̂ is a smoothing transform and ϕt is the Fourier–Stieltjes transform of ρt. Assuming that the initial measure ρ0 belongs to the domain of attraction of a stable law, we describe asymptotic properties of ρt, as t→∞. We consider the boundary regime when the standard normalization leads to a degenerate limit and find an appropriate scaling ensuring a non-degenerate self-similar limit. Our approach is based on a probabilistic representation of probability measures (ρt)t⩾0 that refines the corresponding construction proposed in Bassetti and Ladelli, (2012).

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:2:p:677-693
    DOI: 10.1016/j.spa.2019.03.005
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    Cited by:

    1. 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.
    2. Bassetti, Federico & Ladelli, Lucia, 2023. "Central limit theorem in uniform metrics for generalized Kac equations," Stochastic Processes and their Applications, Elsevier, vol. 166(C).

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