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

Bohdan Maslowski, and Jan Seidler. "Strong Feller solutions to SPDE's are strong Feller in the weak topology." Studia Mathematica 148.2 (2001): 111-129. <http://eudml.org/doc/285225>.

@article{BohdanMaslowski2001,
abstract = {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 σ-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.},
author = {Bohdan Maslowski, Jan Seidler},
journal = {Studia Mathematica},
keywords = {strongly Feller process; stochastic partial differential equation},
language = {eng},
number = {2},
pages = {111-129},
title = {Strong Feller solutions to SPDE's are strong Feller in the weak topology},
url = {http://eudml.org/doc/285225},
volume = {148},
year = {2001},
}

TY - JOUR
AU - Bohdan Maslowski
AU - Jan Seidler
TI - Strong Feller solutions to SPDE's are strong Feller in the weak topology
JO - Studia Mathematica
PY - 2001
VL - 148
IS - 2
SP - 111
EP - 129
AB - 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 σ-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.
LA - eng
KW - strongly Feller process; stochastic partial differential equation
UR - http://eudml.org/doc/285225
ER -