Generalized Ornstein-Uhlenbeck processes in catalytic media

A class of infinite dimensional Ornstein-Uhlenbeck processes that arise as solutions of stochastic partial differential equations with noises generated by measure-valued catalytic processes is investigated.The first topic is the determination of the covariance structure of the processes and identification of the Hilbert space in which the solutions live and have continuous paths. Example of catalysts which are singular measures or measure-valued processes are given and results on the behavior of the corresponding catalytic Ornstein-Uhlenbeck processes are obtained.The second main topic is the study of the special case in which the catalyst is given by a super-Brownian motion. Continuity theorems are established and results on the regularity properties on and off the catalyst are obtained. The third main topic is to prove that the catalytic Ornstein-Uhlenbeck process with super-Brownian catalyst in one dimension arises as a high density fluctuation limit of a super-Brownian motion with a super-Brownian catalyst and with immigration using some of the techniques established before, particularly those based on properties of Sobolev spaces and Laplace functionals