Journal of Applied Mathematics and Stochastic Analysis
Volume 13 (2000), Issue 1, Pages 15-24
Quasi-Feller Markov chains
LAAS-CNRS, 7 A v. du Colonel Roche, Toulouse Cédex 4 31077, France
Received 1 October 1998; Revised 1 October 1999
Copyright © 2000 Jean B. Lasserre. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the class of Markov kernels for which the weak or strong
Feller property fails to hold at some discontinuity set. We provide a simple necessary and sufficient condition for existence of an invariant probability measure as well as a Foster-Lyapunov sufficient condition. We also
characterize a subclass, the quasi (weak or strong) Feller kernels, for which
the sequences of expected occupation measures share the same asymptotic
properties as for (weak or strong) Feller kernels. In particular, it is shown
that the sequences of expected occupation measures of strong and quasi
strong-Feller kernels with an invariant probability measure converge setwise to an invariant measure.