Elliptic operators related to infinite dimensional dissipative systems (Q2759738)

Let \(L\) be the Ornstein-Uhlenbeck generator in a separable real Hilbert space \((H,\langle \cdot, \cdot\rangle)\). The author looks at perturbations of \(L\) of the form NEWLINE\[NEWLINEN_0\varphi= L\varphi+\langle F_0(x), d\varphi\rangle,NEWLINE\]NEWLINE where \(F_0\) is the minimal section of some mapping \(F:D(F)\subset H\to 2^H\). The assumptions on \(F\) include the requirement to approximate \(F_0\) by dissipative Lipschitz-continuous and Gateaux-differentiable functions \(F_\alpha\), \(\alpha>0\). Next, the approximating problem determined by \(F_\alpha\) is investigated, the existence of a unique mild solution of the associated stochastic differential equation is proved and several estimates for the corresponding invariant measures are given. Under slightly stronger assumptions on \(F\) the main result follows: There is a Borel measure \(\nu\) on \(H\) such that \(N:= \overline{N}_0\) is maximal dissipative on \(L^2(H,\nu)\), \(e^{tN}\) is a \(C_0\)-semigroup on \(L^2(H,\nu)\) and \(\nu\) is invariant for \(e^{tN}\).NEWLINENEWLINEFor the entire collection see [Zbl 0968.00044].