Work

Explosion speed of continuous state branching processes indexed by the Esscher transform

Year: 2026

Type: article

Abstract: A branching process Z is said to be non conservative if it hits ∞ in a finite time with positive probability. It is well known that this happens if and only if the branching mechanism φ of Z satisfies ∫0+dλ∕|φ(λ)|<∞. We construct on the same probability space a family of conservative continuous state branching processes Z(ε), ε≥0, each process Z(ε) having φ(ε)(λ)=φ(λ+ε)−φ(ε) as branching mechanism, and such that the family Z(ε), ε≥0 converges a.s. to Z, as ε→0. Then we study the speed of convergence of Z(ε), when ε→0, referred to here as the explosion speed. More specifically, we characterize the functions f with limε→0f(ε)=∞ and such that the first passage times σε=inf{t:Zt(ε)≥f(ε)} converge toward the explosion time of Z. Necessary and sufficient conditions are obtained for the weak convergence and convergence in L1. Then we give a sufficient condition for the almost sure convergence.

Source: Electronic Journal of Probability

DOI: 10.1214/26-ejp1529

Authors: Loïc Chaumont, Clément Lamoureux

Institutions: Centre National de la Recherche Scientifique, Laboratoire Angevin de Recherche en Mathématiques, Université d'Angers

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $675

Funder: Agence Nationale de la Recherche