On the extinction of Continuous State Branching Processes with catastrophes

Vincent Bansaye (1) , Juan Carlos Pardo Millan (2) , Charline Smadi (1, 3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 CIMAT - Centro de Investigación en Matemáticas
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

We consider continuous state branching processes (CSBP) with additional multiplicative jumps modeling dramatic events in a random environment. These jumps are described by a Lévy process with bounded variation paths. We construct a process of this class as the unique solution of a stochastic differential equation. The quenched branching property of the process allows us to derive quenched and annealed results and to observe new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish the probability of extinction. Restricting our attention to the critical and subcritical cases, we show that four regimes arise for the speed of extinction, as in the case of branching processes in random environment in discrete time and space. The proofs are based on the precise asymptotic behavior of exponential functionals of Lévy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection.

Mots clés: Continuous State Branching Processes; Lévy processes; Poisson Point Processes; Stochastic Differential Equation; random environment

Domaines: Probabilités [math.PR]

Vincent Bansaye, Juan Carlos Pardo Millan, Charline Smadi. On the extinction of Continuous State Branching Processes with catastrophes. Electronic Journal of Probability, 2013, 18, pp.1-31. ⟨10.1214/EJP.v18-2774⟩. ⟨hal-00781203v3⟩