The structure of entrance laws for time-inhomogeneous Ornstein-Uhlenbeck Processes with L\'evy Noise in Hilbert spaces

TL;DR

This paper characterizes all entrance laws for time-inhomogeneous Ornstein-Uhlenbeck processes with Lévy noise in Hilbert spaces, extending classical results and providing explicit formulas and applications.

Contribution

It explicitly identifies extremal entrance laws with finite weak first moments and shows their unique representation, generalizing Dynkin's classical results to Lévy noise in Hilbert spaces.

Findings

Explicit formula for Fourier transforms of extremal entrance laws

Unique integral representation of all entrance laws with finite weak first moments

Application to uniqueness of T-periodic entrance laws

Abstract

This paper is about the structure of all entrance laws (in the sense of Dynkin) for time-inhomogeneous Ornstein-Uhlenbeck processes with L\'evy noise in Hilbert state spaces. We identify the extremal entrance laws with finite weak first moments through an explicit formula for their Fourier transforms, generalising corresponding results by Dynkin for Wiener noise and nuclear state spaces. We then prove that an arbitrary entrance law with finite weak first moments can be uniquely represented as an integral over extremals. It is proved that this can be derived from Dynkin's seminal work "Sufficient statistics and extreme points" in Ann. Probab. 1978, which contains a purely measure theoretic generalization of the classical analytic Krein-Milman and Choquet Theorems. As an application, we obtain an easy uniqueness proof for T - periodic entrance laws in the general periodic case. A number of further applications to concrete cases are presented.