Superprocesses with Dependent Spatial Motion and General Branching Densities

Donald Dawson,Hao Wang,Zenghu Li

doi:10.1214/ejp.v6-98

Abstract

We construct a class of superprocesses by taking the high density limit of a sequence of interacting-branching particle systems. The spatial motion of the superprocess is determined by a system of interacting diffusions, the branching density is given by an arbitrary bounded non-negative Borel function, and the superprocess is characterized by a martingale problem as a diffusion process with state space $M({\bf R})$, improving and extending considerably the construction of Wang (1997, 1998). It is then proved in a special case that a suitable rescaled process of the superprocess converges to the usual super Brownian motion. An extension to measure-valued branching catalysts is also discussed.

Highlights

For a given topological space E, let B(E) denote the totality of all bounded Borel functions on E and let C(E) denote its subset comprising of continuous functions
The diffusion process generated by A arises as the high density limit of a sequence of interacting particle systems described by (1.6); see Wang (1997, 1998) and section 4 of this paper
The high density limit of the interacting-branching particle system is considered in section 4, which gives a solution of the martingale problem of the superprocess with dependent spatial motion (SDSM) in the special case where σ ∈ C(R)+ can be extended into a continuous function on R

Summary

Introduction

For a given topological space E, let B(E) denote the totality of all bounded Borel functions on E and let C(E) denote its subset comprising of continuous functions. The diffusion process generated by A arises as the high density limit of a sequence of interacting particle systems described by (1.6); see Wang (1997, 1998) and section 4 of this paper. The high density limit of the interacting-branching particle system is considered, which gives a solution of the martingale problem of the SDSM in the special case where σ ∈ C(R)+ can be extended into a continuous function on R. Where gtm(x, y) denotes the transition density of the m-dimensional standard Brownian motion; see e.g. Friedman (1964, p.24)

Function-valued dual processes

Interacting-branching particle systems

Continuous branching density

Measurable branching density

Rescaled limits

Measure-valued catalysts