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

2018/04/15 by Bogus, Kamil, Buraczewski, Dariusz, Marynych, Alexander
#60F05 #82C40 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1804.05418

Abstract

For a time dependent family of probability measures (ρ_t)_{t≥ 0} we consider a kinetic-type evolution equation ∂ ϕ_t/∂ t + ϕ_t = \widehatQ ϕ_t where \widehatQ is a smoothing transform and ϕ_t is the Fourier--Stieltjes transform of ρ_t. Assuming that the initial measure ρ₀ belongs to the domain of attraction of a stable law, we describe asymptotic properties of ρ_t, as t→∞. We consider the critical 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 (ρ_t)_{t≥ 0} that refines the corresponding construction proposed in Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928--1961, 2012].

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