Stochastic equations, flows and measure-valued processes

Details Stochastic equations, flows and measure-valued processes Stochastic equations, flows and measure-valued processes

2010-09-03

arxiv.org/abs/1009.0578v2

Annals of Probability 2012, Vol. 40, No. 2, 813-857

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP629 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP629

We first prove some general results on pathwise uniqueness, comparison property and existence of nonnegative strong solutions of stochastic equations driven by white noises and Poisson random measures. The results are then used to prove the strong existence of two classes of stochastic flows associated with coalescents with multiple collisions, that is, generalized Fleming--Viot flows and flows of continuous-state branching processes with immigration. One of them unifies the different treatments of three kinds of flows in Bertoin and Le Gall [Ann. Inst. H. Poincar\'{e} Probab. Statist. 41 (2005) 307--333]. Two scaling limit theorems for the generalized Fleming--Viot flows are proved, which lead to sub-critical branching immigration superprocesses. From those theorems we derive easily a generalization of the limit theorem for finite point motions of the flows in Bertoin and Le Gall [Illinois J. Math. 50 (2006) 147--181].

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