Discontinuous Superprocesses with Dependent Spatial Motion

TL;DR

This paper constructs a new class of superprocesses with dependent spatial motion and general branching mechanisms, extending previous models to include discontinuities and more complex interactions, with proofs of existence, uniqueness, and property analysis.

Contribution

It introduces a novel class of discontinuous superprocesses with dependent spatial motion and general branching, expanding the theoretical framework beyond quadratic branching mechanisms.

Findings

Constructed the superprocess as a weak limit of particle systems.

Proved uniqueness using a generalized localization method.

Analyzed properties of the new superprocess.

Abstract

We construct a class of discontinuous superprocesses with dependent spatial motion and general branching mechanism. The process arises as the weak limit of critical interacting-branching particle systems where the spatial motions of the particles are not independent. The main work is to solve the martingale problem. When we turn to the uniqueness of the process, we generalize the localization method introduced by [D.W. Stroock, Diffusion processes associated with Levy generators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32(1975) 209--244] to the measure-valued context. As for existence, we use particle system approximation and a perturbation method. This work generalizes the model introduced in [D.A. Dawson, Z. Li, H. Wang, Superprocesses with dependent spatial motion and general branching densities, Electron. J. Probab. 6(2001), no.25, 33 pp. (electronic)] where quadratic branching mechanism was considered. We also investigate some properties of the process.