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On the extinction of continuous state branching processes with catastrophes


 
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1. Title Title of document On the extinction of continuous state branching processes with catastrophes
 
2. Creator Author's name, affiliation, country Vincent Bansaye; École Polytechnique
 
2. Creator Author's name, affiliation, country Juan Carlos Pardo Millan; CIMAT; Mexico
 
2. Creator Author's name, affiliation, country Charline Smadi; École Polytechnique and Université Paris-Est; France
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Continuous State Branching Processes, Lévy processes, Poisson Point Processes, Random Environment, Extinction, Long time behavior
 
3. Subject Subject classification 60J80, 60J25, 60G51, 60H10, 60G55, 60K37.
 
4. Description 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 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.
 
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7. Date (YYYY-MM-DD) 2013-12-18
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/2774
 
10. Identifier Digital Object Identifier 10.1214/EJP.v18-2774
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 18
 
12. Language English=en en
 
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