Selfadjointness of some infinite-dimensional elliptic operators and application to stochastic quantization
Let \(H\) be a separable Hilbert space. The authors consider the following linear operator \[ K\varphi =L\varphi-\langle C^{1/2}DU(x),C^{1/2}D\varphi\rangle; \qquad {\mathcal D}(K) = {\mathcal D}(L) \cap C^2_b(H), \] where \(D\varphi\) is Fréchet derivative of \(\varphi\), \(L\varphi = {1\over 2} \operatorname {Tr}[CD^2\varphi]+ \langle Ax,D\varphi\rangle\) is the Ornstein-Uhlenbeck operator, \(A<0\) is linear operators on \(H\), \(C=(-A)^{-\varepsilon}\), \(U\) is a mapping from \(H\) to \({\mathbb R}\) such that \(e^{-U}\in L^p(H,\mu), p\geq 1\). If \(U\in W^{1,4}(H,\mu)\), and \((-A)^{-1-\varepsilon}\) is of trace class, it is proved that operator \(K\) is essentally selfadjoint in the space \(L^2(H,\nu)\), where \(\mu\) is the invariant measure associated with \(L\), \(\nu\) is the probability measure \(\nu(dx)=\operatorname {const}e^{-2U(x)} dx\). An application to stochastic quantization is given.
scientific article; zbMATH DE number 1547918