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

Solutions to kinetic-type evolution equations: beyond the boundary case

“…The latter follows from Lemma 3.6 in [9], but can also be directly inferred as a simple consequence of the fact that ϑ ∈ int D Φ . Moreover, from Lemma 3.9 in [9], we infer the existence of some δ > 0 such that…”

“…In the case where µ 0 is in the domain of normal attraction of a γ-stable law (and if µ 0 is additionally centered when γ > 1), the asymptotic behavior of µ t as t → ∞ was found in [1] for γ < ϑ and in [9] in the boundary case γ = ϑ. In this note, we address the remaining case γ > ϑ or, more generally, the case where…”

“…The corresponding family of Fourier transforms will be denoted by (φ t ) t≥0 . The purpose of this note is to demonstrate how, in a regime that has not been addressed before, the asymptotic behavior of µ t (in the weak sense) as t → ∞ (equivalently, the asymptotic behavior of φ t as t → ∞) can be derived from recent progress on kinetic-type equations [9] and on the extrema of branching random walks [10,13].…”

“…for some p > ϑ. If ϑ ≥ 2, then µ 0 has a finite second moment and the asymptotic behavior of µ t can be deduced from [1] if ϑ > 2 and from [9] if ϑ = 2. Therefore, we shall from now on suppose that (1.4) holds and that ϑ < 2.…”

“…Representation of solutions: the branching random walk connection. 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 [9], see also [1] and the references therein for earlier results.…”