Strong Feller solutions to SPDE's are strong Feller in the weak topology

Bohdan Maslowski¹ ; Jan Seidler¹

¹Mathematical Institute Academy of Sciences Žitná 25 115 67 Praha 1, Czech Republic

Studia Mathematica, Tome 148 (2001) no. 2, pp. 111-129

Résumé

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.

Détail

DOI : 10.4064/sm148-2-2

Keywords: wide class markov processes hilbert space defined semilinear stochastic partial differential equations their transition semigroups map bounded borel functions functions weakly continuous bounded sets provided map bounded borel functions functions continuous norm topology particular ornstein uhlenbeck process strong feller norm topology only strong feller bounded weak topology consequence possible strengthen results long time behaviour strongly feller processes extend embedded markov chains method constructing sigma finite invariant measure replacing recurrent compact sets recurrent balls transient prove exit time every weakly compact set finite almost surely