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

Kamil Bogus, Dariusz Buraczewski and Alexander Marynych

Stochastic Processes and their Applications, 2020, vol. 130, issue 2, 677-693

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).

Keywords: Biggins martingale; Derivative martingale; Kac model; Kinetic equation; Random trees; Smoothing transform (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (2)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:130:y:2020:i:2:p:677-693

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DOI: 10.1016/j.spa.2019.03.005

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