Interacting Superprocesses with Discontinuous Spatial Motion and their Associated SPDEs

A class of interacting superprocesses arising from branching particle systems with continuous spatial motions, called superprocesses with dependent spatial motion (SDSMs), has been introduced and studied in Wang [26] and Dawson et al. [8]. In this paper, we extend the model to allow discontinuous spatial motions. Under Lipschitz condition for coefficients, we show that under a proper rescaling, branching particle systems with jump-diffusion underlying motions in a random medium converge to a measure-valued process, called stable SDSM. We further characterize this stable SDSM as a unique solution of a well-posed martingale problem. To prove the uniqueness of the martingale problem, we establish the C2+γ-regularity for the transition semigroup of a class of jumpdiffusion processes, which may be of independent interest. The corresponding stochastic partial differential equations (SPDE in abbreviation) for each stable SDSM is also derived in this paper. AMS Subject Classifications (2000): Primary 60J80, 60G57, 60K35, 60K37; Secondary 60G52