On the Ornstein-Uhlenbeck operator in convex sets of Banach spaces

We study the Ornstein-Uhlenbeck operator and the Ornstein-Uhlenbeck semigroup in an open convex subset of an infinite dimensional separable Banach space $X$. This is done by finite dimensional approximation. In particular we prove Logarithmic-Sobolev and Poincar'e inequalities, and thanks to these inequalities we deduce the spectral properties of the Ornstein-Uhlenbeck operator.