Feller Markov Chains

Abstract

In this chapter we consider the class of MCs on a LCS metric space X that have either the weak-or the strong-Feller property introduced in ¡ì4.4 (see Definition 4.4.2). The Feller property is a continuity property on the t.p.f. P of the MC. In particular, it permits to derive simple necessary and/or sufficient conditions for existence of an invariant p.m. for P (recall that most results in the previous chapters assumed that the MC had an invariant p.m.). In fact, most conditions for existence of an invariant p.m. do assume the weak-Feller property, and as can be shown in simple examples, the failure to satisfy this continuity condition can have important consequences (see, e.g., the MC defined by (7.3.1) below).