Abstract
We consider continuous state branching processes (CSBP's) with additional multiplicative jumps modeling dramatic events in a random environment. These jumps are described by a Levy process with bounded variation paths. We construct the associated class of processes 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 make appear new asymptotic behaviors. We characterize the Laplace exponent of the process as the solution of a backward ordinary differential equation and establish when it becomes extinct. For a class of processes for which extinction and absorption coincide (including the $\alpha$ stable CSBP's plus a drift), we determine the speed of extinction. Four regimes appear, as in the case of branching processes in random environment in discrete time and space.The proofs rely on a fine study of the asymptotic behavior of exponential functionals of Levy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection.
Highlights
Continuous state branching processes (CSBP) are the analogues of Galton-Watson (GW) processes in continuous time and continuous state space
We focus on the case when the branching mechanism associated to the CSBP with catastrophes Y has the form ψ(λ) = −gλ + cλ1+β, for β ∈
We prove that the speed of extinction is directly related to the asymptotic behavior of exponential functionals of Levy processes
Summary
Continuous state branching processes (CSBP) are the analogues of Galton-Watson (GW) processes in continuous time and continuous state space. The stable case with drift, i.e. ψ(λ) = −gλ + cλ1+β, with β in It is of special interest in this paper since the Laplace exponent can be computed explicitly and it can be used to derive asymptotic results for more general cases. We show that conditionally on the times and the effects of the catastrophes, the process satisfies the branching property It yields a particular class of CSBP in random environment, which can be obtained as the scaling limit of GW processes in random environment (see [4]). This result is the key to deducing the different extinction regimes.
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