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

A continuous-time particle system on the real line verifying the branching property and an exponential integrability condition is called a branching Lévy process, and its law is characterized by a triplet $(\sigma^2,a,\Lambda)$. We obtain a necessary and sufficient condition for the convergence of the derivative martingale of such a process to a non-trivial limit in terms of $(\sigma^2,a,\Lambda)$. This extends previously known results on branching Brownian motions and branching random walks. To obtain this result, we rely on the spinal decomposition and establish a novel zero-one law on the perpetual integrals of centred Lévy processes conditioned to stay positive.

Submitted to Bernoulli A necessary and sufficient condition for the convergence of the derivative martingale in a branching Lévy process arXiv:2105.07919v2 [math.PR] 24 Feb 2022 BASTIEN MALLEIN1 and QUAN SHI2,* 1 Université Sorbonne Paris Nord, LAGA, UMR 7539, F-93430, Villetaneuse, France,…

related reading

• On the Functional Levy-Ito Stochastic Calculusarxiv.org
• Functional Equations for the Stochastic Exponentialarxiv.org
• A Primer on Stochastic Partial Differential Equationsmath.utah.edu
• The Concentration of Invariant Measures for Stochastic Dynamical Systems with Locally Lipschitz Continuous Coefficients in R^darxiv.org
• Density Dependent Singular Stochastic Differential Equationsarxiv.org
• Advanced stochastic analysishairer.org
• Dynamical Approach to RMT.pdfpeople.math.harvard.edu
• Research • Leonid Petrov. Integrable Probabilitylpetrov.cc
• An optimization perspective on log-concave sampling and beyond | Sinho Chewichewisinho.github.io
• 0907.4178arxiv.org
• Comparison principle for stochastic heat equation on Rdmath.emory.edu
• Levy Processes and Levy White Noise as Tempered Distributionsarxiv.org