Differentiable perturbations of Ornstein-Uhlenbeck operators

We prove an extension theorem for a small perturbation of the Ornstein-Uhlenbeck operator (L,D(L)) in the space of all uniformly continuous and bounded functions f : H → R, where H is a separable Hilbert space. We consider a perturbation of the form N0φ = Lφ + 〈Dφ,F 〉 where F : H → H is bounded and Fréchet differentiable with uniformly continuous and bounded differential. Hence, we prove that N0 is m-dissipative and its closure in Cb(H) coincides with the infinitesimal generator of a diffusion semigroup associated to a stochastic differential equation in H.