First order perturbations of Dirichlet operators: Existence and uniqueness
The author studies perturbations of the type \(B\cdot\nabla\) of Dirichlet operators \(L^0\) associated with Dirichlet forms of the type \({\mathcal E}^0(u,v)= {1\over 2}\int\langle\nabla u,\nabla v\rangle_H d\mu\) on \(L^2(E,H)\). Here, \(E\) is a finite- or infinite-dimensional Banach space, \(H\) is a Hilbert space densely and continuously embedded in \(E\), and \(B: E\to H\) is a vector field in \(L^3(E,H; \mu)\). It is shown that if \(B\) satisfies \[ \int\langle B,\nabla u\rangle_H d\mu\leq 0 \] for all \(u\geq 0\) and if \({\mathcal E}^0\) is quasi-regular, then there always exists a closed extension of \(Lu= L^0u+ \langle B,\nabla u\rangle_H\) that generates a \(C_0\)-semigroup \(T_t\) of contractions which is sub-Markovian; that is, \(0\leq u\leq 1\) implies \(0\leq T_tu\leq 1\) for all \(t\geq 0\). If \(D\) is an appropriate core for \(L^0\), it is proved that there is only one closed extension of \((L,D)\) in \(L^2(E,\mu)\) generating a strongly continuous semigroup. The result is applied to the operator \(\Delta\mu+ B\cdot\nabla\), where \(\Delta_H\) is the Gross-Laplacian on an abstract Wiener space and \(B= -\text{id}_E+ v\), where \(v\) takes values in the Cameron-Martin space \(H\).
scientific article; zbMATH DE number 943137