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

Abstract

For a time dependent family of probability measures (ρt) t 0 we consider a kinetic-type evolution equation ∂φt/∂t + φt = Qφt where Q 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 [Ann.

Self-similar solutions to kinetic-type evolution equations: beyond the boundary case

“…Moreover, the skeleton branching random walk (Z nδ ) n∈N0 is non-lattice by (A1) and satisfies E u∈I δ V (u) 2 e −V (u) ∈ (0, ∞). (3.3) The latter follows from Lemma 3.6 in [12]. As a corollary of Lemma 2.2, we get the following: Proposition 3.1.…”

“…In both cases, if φ 0 satisfies (1.8) with γ > ϑ, then µ 0 ∈ M 1 p (R) for some ϑ < p ≤ 2. In this more general situation, we demonstrate how the asymptotic behavior of µ t can be derived from recent progress on kinetic-type equations [12] and the extrema of branching random walks [17,21]. Our proof works under a mild X log Xtype moment condition (cf.…”

“…In the recent work [12] the authors establish a connection with continuous-time branching processes which enables them to treat the boundary case γ = ϑ in which…”

“…Given an initial distribution µ 0 and the random vector A, we give a representation of µ t as the law of a continuous-time branching random walk at time t. The exact form of representation was developed in [12], see also [4] and the references therein for earlier results.…”

A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process

“…[AS14] for a proof in the context of branching random walks and [BBM20] for a continuous-time extension). When Z t converges to a non-degenerate limit Z ∞ , the random variable Z ∞ is related to the maximal displacement of the branching process.…”