Gauss-Markov processes on Hilbert spaces

TL;DR

This paper extends Itô's characterization of zero-mean stationary Gauss-Markov processes to Hilbert spaces, identifying them as solutions to linear stochastic differential equations with singular coefficients and characterizing their transition semigroup and generator.

Contribution

It introduces a novel infinite-dimensional framework for Gauss-Markov processes on Hilbert spaces, extending classical results to this setting.

Findings

Characterization of Gauss-Markov processes as solutions to SDEs with singular coefficients

Identification of transition semigroup and generator in Hilbert space setting

Extension of Courr ext{è}ge's classical result to infinite dimensions

Abstract

K. It\^{o} characterised in \cite{ito} zero-mean stationary Gauss Markov-processes evolving on a class of infinite-dimensional spaces. In this work we extend the work of It\^{o} in the case of Hilbert spaces: Gauss-Markov families that are time-homogenous are identified as solutions to linear stochastic differential equations with singular coefficients. Choosing an appropriate locally convex topology on the space of weakly sequentially continuous functions we also characterize the transition semigroup, the generator and its core thus providing an infinite-dimensional extension of the classical result of Courr\`ege \cite{courrege} in the case of Gauss-Markov semigroups.