Ornstein–Uhlenbeck and renormalization semigroups

The Ornstein–Uhlenbeck semigroup combines Gaussian diffusion with the flow of a linear vector field. In infinite-dimensional settings there can be non-Gaussian invariant measures. This gives a context for one version of the renormalization group. The adjoint of the Ornstein–Uhlenbeck 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.