Continuous local time of a purely atomic immigration superprocess with dependent spatial motion

TL;DR

This paper constructs a new class of immigration superprocesses with dependent spatial motion, proves the existence of a Holder continuous local time, and establishes scaling limit theorems for these processes and their local times.

Contribution

It introduces a novel superprocess model driven by Poisson processes, extending previous work, and analyzes its local time properties and scaling limits.

Findings

Superprocess with dependent spatial motion constructed as a strong solution.

Local time exists and is Holder continuous of order less than 1/2.

Scaling limits for the superprocess and its local time are established.

Abstract

A purely atomic immigration superprocess with dependent spatial motion in the space of tempered measures is constructed as the unique strong solution of a stochastic integral equation driven by Poisson processes based on the excursion law of a Feller branching diffusion, which generalizes the work of Dawson and Li (2003). As an application of the stochastic equation, it is proved that the superprocess possesses a local time which is Holder continuous of order $\alpha$ for every $\alpha< 1/2$. We establish two scaling limit theorems for the immigration superprocess, from which we derive scaling limits for the corresponding local time.