Cylindrical Levy processes in Banach spaces

TL;DR

This paper develops a theoretical framework for cylindrical Levy processes in Banach spaces, including their decompositions, stochastic integrals, and invariant measures, advancing the understanding of infinite-dimensional stochastic processes.

Contribution

It introduces a comprehensive analysis of cylindrical Levy processes, including their Levy-Ito decompositions, series representations, and construction of cylindrical Ornstein-Uhlenbeck processes.

Findings

Cylindrical Levy processes admit Levy-Ito decompositions.

Series representations enable construction of cylindrical Ornstein-Uhlenbeck processes.

Invariant measures for these processes are characterized.

Abstract

Cylindrical probability measures are finitely additive measures on Banach spaces that have sigma-additive projections to Euclidean spaces of all dimensions. They are naturally associated to notions of weak (cylindrical) random variable and hence weak (cylindrical) stochastic processes. In this paper we focus on cylindrical Levy processes. These have (weak) Levy-Ito decompositions and an associated Levy-Khintchine formula. If the process is weakly square integrable, its covariance operator can be used to construct a reproducing kernel Hilbert space in which the process has a decomposition as an infinite series built from a sequence of uncorrelated bona fide one-dimensional Levy processes. This series is used to define cylindrical stochastic integrals from which cylindrical Ornstein-Uhlenbeck processes may be constructed as unique solutions of the associated Cauchy problem. We demonstrate that such processes are cylindrical Markov processes and study their (cylindrical) invariant measures.