Yaglom limit theorem for a critical continuous-state branching process in Lévy environment: special cases

M. González Velasco, P. Martín Chávez, J. C. Pardo, I. Puerto García

In this talk, we consider continuous-state branching processes evolving in a Lévy random environment, which extend classical continuous-state branching processes by incorporating environmental fluctuations. They can also be viewed as scaling limits of Galton–Watson processes in random environments. Our goal is to investigate the asymptotic behaviour of such processes in the critical regime. More precisely, under a Spitzer-type condition on the associated Lévy environment, we prove the existence of a Yaglom limit, that is, a non-degenerate limiting distribution for the process conditioned on survival. In the case where the environment belongs to the domain of attraction of a stable law, a more precise characterization can be obtained in terms of exponential functionals of Lévy processes. Moreover, in the particular case of a Brownian environment, we derive an explicit description of the limiting distribution.

Keywords: branching processes, Lévy processes, limit distributions

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