The catalytic Ornstein-Uhlenbeck process with superprocess catalyst

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.