On the domain of non-symmetric and, possibly, degenerate Ornstein–Uhlenbeck operators in separable Banach spaces

Abstract

Let X be a separable Banach space and let X^* be its topological dual. Let Q:X^*\rightarrow X be a linear, bounded, non-negative, and symmetric operator and let A:D(A)\subseteq X\rightarrow X be the infinitesimal generator of a strongly continuous semigroup of contractions on X . We consider the abstract Wiener space (X,\mu_\infty,H_\infty) , where \mu_\infty is a centered non-degenerate Gaussian measure on X with covariance operator defined, at least formally, as Q_\infty=\int_0^{+\infty} e^{sA}Qe^{sA^*}\,ds, and H_\infty is the Cameron–Martin space associated to \mu_\infty . Let H be the reproducing kernel Hilbert space associated with Q with inner product [\cdot,\cdot]_H . We assume that the operator Q_\infty A^*:D(A^*)\subseteq X^*\rightarrow X extends to a bounded linear operator B\in\mathcal{L}(H) which satisfies B+B^*=-\mathrm{Id}_H , where \mathrm{Id}_H denotes the identity operator on H . Let D and D^2 be the first and second order Fréchet derivative operators. We denote by D_H and (D_H,D^2_H) the closure in L^2(X,\mu_\infty) of the operators QD and (QD,QD^2) , respectively, defined on smooth cylindrical functions, and by W^{1,2}_H(X,\mu_\infty) and W^{2,2}_H(X,\mu_\infty) , respectively, their domains in L^2(X,\mu_\infty) . Furthermore, we denote by D_{A_\infty} the closure of the operator Q_\infty A^*D in L^2(X,\mu_\infty) defined on smooth cylindrical functions, and by W^{1,2}_{A_\infty}(X,\mu_\infty) the domain of D_{A_\infty} in L^2(X,\mu_\infty) . We characterize the domain of the operator L , associated to the bilinear form (u,v)\mapsto-\int_{X}[BD_Hu,D_Hv]_H\,d\mu_\infty, \quad u,v\in W^{1,2}_H(X,\mu_\infty), in L^2(X,\mu_\infty) . More precisely, we prove that D(L) coincides, up to an equivalent renorming, with a subspace of W^{2,2}_H(X,\mu_\infty)\cap W^{1,2}_{A_\infty}(X,\mu_\infty) . We stress that we are able to treat the case when L is degenerate and non-symmetric.