976 research outputs found

    Stochastic equations, flows and measure-valued processes

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

    Asymptotic behavior of the Poisson--Dirichlet distribution for large mutation rate

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    The large deviation principle is established for the Poisson--Dirichlet distribution when the parameter θ\theta approaches infinity. The result is then used to study the asymptotic behavior of the homozygosity and the Poisson--Dirichlet distribution with selection. A phase transition occurs depending on the growth rate of the selection intensity. If the selection intensity grows sublinearly in θ\theta, then the large deviation rate function is the same as the neutral model; if the selection intensity grows at a linear or greater rate in θ\theta, then the large deviation rate function includes an additional term coming from selection. The application of these results to the heterozygote advantage model provides an alternate proof of one of Gillespie's conjectures in [Theoret. Popul. Biol. 55 145--156].Comment: Published at http://dx.doi.org/10.1214/105051605000000818 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Tanaka formula and local time for a class of interacting branching measure-valued diffusions

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    We construct superprocesses with dependent spatial motion (SDSMs) in Euclidean spaces and show that, even when they start at some unbounded initial positive Radon measure such as Lebesgue measure on Rd\R^d, their local times exist when d3d\le3. A Tanaka formula is also derived