On the Parabolic Martin Boundary of the Ornstein-Uhlenbeck Operator on Wiener Space

Abstract

We study the positive parabolic functions of the Ornstein-Uhlenbeck operator on an abstract Wiener space $E$ using the approach developed by Dynkin. This involves first proving a characterization of the entrance space of the corresponding Ornstein-Uhlenbeck semigroup and deriving an integral representation for an arbitrary entrance law in terms of extreme ones. It is shown that the Cameron-Martin densities are extreme parabolic functions, but that if $\dim E = \infty$, not every positive parabolic function has an integral representation in terms of those (which is in contrast to the finite-dimensional case). Furthermore, conditions for a parabolic function to be representable in terms of Cameron-Martin densities are proved.