Stochastic equations, flows and measure-valued processes

TL;DR

This paper establishes foundational results on stochastic equations driven by white noise and Poisson measures, and applies them to prove the existence and scaling limits of stochastic flows related to coalescents, branching processes, and superprocesses.

Contribution

It introduces new results on pathwise uniqueness and existence of solutions, and unifies treatments of stochastic flows associated with coalescents and branching processes.

Findings

Proved strong existence of generalized Fleming--Viot flows.

Established scaling limit theorems for these flows.

Unified different types of stochastic flows in a single framework.

Abstract

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].