Electronic Journal of Probability

On the extinction of continuous state branching processes with catastrophes

Vincent Bansaye, Juan Carlos Pardo Millan, and Charline Smadi

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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 Lévy 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 Lévy processes. Finally, we apply these results to a cell infection model and determine the mean speed of propagation of the infection.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 106, 31 pp.

Accepted: 18 December 2013
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60G51: Processes with independent increments; Lévy processes 60H10: Stochastic ordinary differential equations [See also 34F05] 60G55: Point processes 60K37: Processes in random environments

Continuous State Branching Processes Lévy processes Poisson Point Processes Random Environment Extinction Long time behavior

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Bansaye, Vincent; Pardo Millan, Juan Carlos; Smadi, Charline. On the extinction of continuous state branching processes with catastrophes. Electron. J. Probab. 18 (2013), paper no. 106, 31 pp. doi:10.1214/EJP.v18-2774. https://projecteuclid.org/euclid.ejp/1465064331

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