On a functional analytic approach for transition semigroups on$L^2(\mu)$

Abstract

By using only analytic tools we prove the positivity of the transitionsemigroup associated formally with the stochastic differential equation$$dX(t)=(AX(t)+F(X(t)))dt+Q^{\frac{1}{2}}dW(t),\,X(0)=x,t\ge 0,\,x\in H$$in the case where $F\in UCB(H,H)$.As a consequence we obtain the existence of an invariant measure of the abovestochastic equation.