Ornstein—Uhlenbeck and Renormalization Semigroups

Abstract

The Ornstein-Uhlenbeck semigroup combines Gaussian diusion with the ow of a linear vector eld. In innite dimensional settings there can be non-Gaussian invariant measures. This gives a context for one version of the renormalization group. The adjoint of the OrnsteinUhlenbeck semigroup with respect to an invariant measure need not be an Ornstein-Uhlenbeck semigroup. This adjoint is the appropriate semigroup to analyze the local stability of the invariant measure under the renormalization group. 2000 Math. Subject Class. Primary 81T17, 82B28, 60G60, 47D06; Secondary 60J60.