Strong Feller solutions to SPDE's are strong Feller in the weak topology
Tom 148 / 2001
For a wide class of Markov processes on a Hilbert space $H$, defined by semilinear stochastic partial differential equations, we show that their transition semigroups map bounded Borel functions to functions weakly continuous on bounded sets, provided they map bounded Borel functions into functions continuous in the norm topology. In particular, an Ornstein–Uhlenbeck process in $H$ is strong Feller in the norm topology if and only if it is strong Feller in the bounded weak topology. As a consequence, it is possible to strengthen results on the long-time behaviour of strongly Feller processes on $H$: we extend the embedded Markov chains method of constructing a $\sigma $-finite invariant measure by replacing recurrent compact sets with recurrent balls, and in the transient case we prove that the last exit time from every weakly compact set is finite almost surely.