Dirichlet operators for dissipative gradient systems (Q2708588)

Denote by \(L\) the Ornstein-Uhlenbeck operator \(L\varphi=\frac{1}{2}\text{Tr}[D^{2}\varphi]+ \langle Ax,D\varphi\rangle\) where \(\varphi\) is a continuous bounded real function defined on a Hilbert space and \(A\) is a selfadjoint strictly negative operator in \(D(A)\) such that \(A^{-1}\) is of trace class and \(\langle Ax,x\rangle\leq -\omega|x|^{2}\), \(x\in H\), \(\omega>0\). Take a convex proper lower continuous mapping \(U:H\rightarrow[0,+\infty]\) and consider the operator \(N_{0}\varphi=L\varphi-\langle \partial U,D\varphi\rangle\), where \(\partial U\) is the subdifferential of \(U\) which can be multi-valued. The author proves that \(N_{0}\) is essentially self-adjoint in the space \(L^{2}(H,\nu)\), where NEWLINE\[NEWLINE \nu(dx)=\frac{e^{-2U(x)}}{\int_{H}e^{-2U(y)}\mu(dy)}\mu(dx) NEWLINE\]NEWLINE and \(\mu\) is the Gaussian measure on \(H\) with mean \(0\) and covariance operator \(Q=-\frac{1}{2}A^{-1}\). Moreover, let \(N\) be the closure of \(N_{0}\) and \(P_{t}=e^{tN}\) be the associated semigroup. The author characterizes the domain of \(N\) and the asymptotic properties of the semigroup (ergodicity, strong mixing, spectral gap). The paper is a generalisation of another one of the author [Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 1, No. 3, 501-519 (1998; Zbl 0931.47023)]. \(N\) is associated with the differential stochastic inclusion NEWLINE\[NEWLINE dX-(AX-\partial U(X)) dt-dW(t)\ni 0,\quad X(0)=x, NEWLINE\]NEWLINE \(W\) being a Brownian motion on \(H\).NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].