Gauss-Markov processes on Hilbert spaces

Ben Goldys, Szymon Peszat and Jerzy Zabczyk

Abstract

K. Itô characterized in [13] zero-mean stationary Gauss Markov-processes evolving on a class of infinite-dimensional spaces. In this work we extend the work of Itô 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 Courrège [3] in the case of Gauss-Markov semigroups.

Keywords: Gauss–Markov process, Ornstein–Uhlenbeck process, Gaussian measure, bw-topology, strict topology, generator.

AMS Subject Classification: Primary 60G15; secondary 60H15, 60J99.