Transition semigroups corresponding to Lipschitz dissipative systems.
Consider the following stochastic differential equation on a separable Hilbert space \(H\): \[ d X_t = (A X_t +F(X_t))d t + dW_t, \] where \(W_t\) is the cylindrical Wiener process, \((A, D(A))\) the generator of a \(C_0\)-semigroup in \(H\) of negative type, and \(F\) a Lipschitz dissipative mapping on \(H\). Let \(P_t\) be the transition semigroup associated to the mild solution which has a unique invariant measure \(\nu\). Then under certain conditions, the infinitesimal generator \(N_p\) of \(P_t\) on \(L^p(\nu)\), \(p\geq 1,\) coincides with the \(L^p\)-closure of the following operator defined on the space of exponential functions: \[ N_0 f(x):= \frac 1 2 \text{Tr}(D^2 f(x)) +\langle x,A^*Df(x)\rangle +\langle F(x), Df(x)\rangle,\;\;x\in H. \] Moreover, it is proved that \(\nu\) is absolutely continuous with respect to the invariant measure associated to the equation with \(F=0.\)
scientific article; zbMATH DE number 2077713