First Order Perturbations of Dirichlet Operators: Existence and Uniqueness

We study perturbations of typeB·∇ of Dirichlet operators (L0, D(L0)) associated with Dirichlet forms of typeE0(u, v)=12∫〈∇u, ∇v〉HdμonL2(E, μ) whereEis a finite or infinite dimensional Banach space. HereHdenotes a Hilbert space densely and continuously embedded inE. Assuming quasi-regularity of (E0, D(E0)) we show that there always exists a closed extension ofLu:=L0u+〈B, ∇u〉Hthat generates a sub-MarkovianC0-semigroup of contractions onL2(E, μ) (resp.L1(E, μ)), ifB∈L2(E; H, μ) and ∫〈B, ∇u〉Hdμ⩽0,u⩾0. IfDis an appropriate core for (L0, D(L0)) we show that there is only one closed extension of (L, D) inL1(E, μ) generating a strongly continuous semigroup. In particular we apply our results to operators of typeΔH+B·∇, whereΔHdenotes the Gross–Laplacian on an abstract Wiener space (E, H, γ) andB=−idE+v, wherevtakes values in the Cameron–Martin spaceH.