The catalytic Ornstein-Uhlenbeck process with superprocess catalyst

TL;DR

This paper investigates a class of catalytic Ornstein-Uhlenbeck processes driven by measure-valued superprocess catalysts, providing an affine characterization and analyzing their properties in both quenched and annealed frameworks.

Contribution

It introduces a novel class of catalytic Ornstein-Uhlenbeck processes with superprocess catalysts and characterizes their affine structure and fundamental properties.

Findings

Affine characterization of the joint process and catalyst

Identification of quenched and annealed process properties

Connection to super-Brownian motion and affine processes

Abstract

The main objective of this work is to study a natural class of catalytic Ornstein-Uhlenbeck (O-U) processes with a measure-valued random catalyst, for example, super-Brownian motion. We relate this to the class of affine processes that provides a unified setting in which to view Ornstein-Uhlenbeck processes, superprocesses, and Ornstein-Uhlenbeck processes with superprocess catalyst. We then review some basic properties of super-Brownian motion which we need and introduce the Ornstein-Uhlenbeck process with catalyst given by a superprocess. The main results are the affine characterization of the characteristic functional-Laplace transform of the joint catalytic O-U process and catalyst process and the identification of basic properties of the quenched and annealed versions of these processes.