Existence and uniqueness of invariant measures for a class of transition semigroups on Hilbert spaces
This paper investigates the following semilinear stochastic differential equation on a separable Hilbert space: \[ d X(t)= (AX(t) + (-A)^{1/2} F(X(t))) d t + d W_t, \] where \(A\) is a negatively definite selfadjoint operator with compact resolvent, \(W\) is the cylindrical Brownian motion, and \(F\) is a global Lipschitz continuous function. It is proved that, if \[ \int_0^1s^{-\varepsilon}\|e^{sA}\|_{HS}^2 \,ds <\infty \] holds for some \(\varepsilon\in (0,1),\) then the equation has a unique mild solution and the corresponding semigroup is strong Feller. If, furthermore, \(\langle x-y, F(x)-F(y)\rangle\) holds for all \(x,y\), then the semigroup has a unique invariant probability measure. These extend some known results for equations without \((-A)^{1/2}\) in front of \(F\).
scientific article; zbMATH DE number 5553631