Generalized self-intersection local time for a superprocess over a stochastic flow

TL;DR

This paper establishes the existence of a generalized self-intersection local time for superprocesses over stochastic flows in low dimensions, extending moment calculation methods to handle dependent spatial motions.

Contribution

It introduces a new approach to prove existence of self-intersection local time for superprocesses with dependent motion, overcoming limitations of previous methods.

Findings

Existence of self-intersection local time in dimensions d ≤ 3.

Extension of moment calculation methods to higher moments.

Development of a Tanaka-like representation for the superprocess.

Abstract

This paper examines the existence of the self-intersection local time for a superprocess over a stochastic flow in dimensions $d\leq3$, which through constructive methods, results in a Tanaka-like representation. The superprocess over a stochastic flow is a superprocess with dependent spatial motion, and thus Dynkin's proof of existence, which requires multiplicity of the log-Laplace functional, no longer applies. Skoulakis and Adler's method of calculating moments is extended to higher moments, from which existence follows.