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 essentially m-dissipative and its closure in C_b(H) coincides with the infinitesimal generator of a diffusion semigroup associated to a stochastic differential equation in H.