Infinite Dimensional Ornstein-Uhlenbeck Processes Driven by Levy Processes

TL;DR

This paper reviews the properties of infinite-dimensional Ornstein-Uhlenbeck processes driven by Lévy processes, highlighting their applications in stochastic PDEs, branching processes, and operator self-decomposable distributions.

Contribution

It provides a comprehensive overview of the probabilistic characteristics and various contexts of these processes, including generalizations to cylindrical noise.

Findings

Analysis of Ornstein-Uhlenbeck processes in Hilbert spaces

Connections to stochastic PDEs and branching processes

Extensions to cylindrical Lévy noise

Abstract

We review the probabilistic properties of Ornstein-Uhlenbeck processes in Hilbert spaces driven by L\'{e}vy processes. The emphasis is on the different contexts in which these processes arise, such as stochastic partial differential equations, continuous-state branching processes, generalised Mehler semigroups and operator self-decomposable distributions. We also examine generalisations to the case where the driving noise is cylindrical.