Regular densities of invariant measures in Hilbert spaces (Q1893839)

Let \(H\) be a separable Hilbert space, \(A\) be the infinitesimal generator of a \(C_0\) semigroup \(S(t)\), \(t \geq 0\), in \(H\), such that for some \(M, \omega > 0\), \(|S(t) |\leq Me^{- \omega t}\) for all \(t \geq 0\), and let \(Q\) be a positive symmetric bounded operator in \(H\) such that the operator \(x \mapsto \int^t_0 S(t) QS^* (s) xds\) is nuclear and its trace is bounded in \(t\). It is well-known that the linear equation \(dZ = AZdt + \sqrt QdW\), \(Z(0) = x \in H\), where \(W\) is a cylindrical Wiener process in \(H\), has a mild solution and there exists a unique invariant measure \(\mu_0\) for \(Z\). The authors consider the perturbed equation \(dX = (AX + F(X)) dt + \sqrt Q dW\), \(X(0) = x\), where \(F\) is a bounded Lipschitz mapping from \(H\) into \(H\), and give conditions under which there exists an invariant measure \(\mu_F\) for \(X\), which is absolutely continuous with respect to \(\mu_0\), and \(d \mu_F/d \mu_0 \in L^2 (H, \mu_0)\). Moreover, under some additional assumptions this density is proved to belong to some Sobolev spaces.