On iterates of strong Feller operators on ordered phase spaces
Volume 101 / 2004
Abstract
Let $(X, {\rm d} )$ be a metric space where all closed balls are compact, with a fixed $\sigma $-finite Borel measure $\mu $. Assume further that $X $ is endowed with a linear order $\preceq $. Given a Markov (regular) operator $P: L^1(\mu ) \to L^1(\mu )$ we discuss the asymptotic behaviour of the iterates $P^n$. The paper deals with operators $P$ which are Feller and such that the $\mu $-absolutely continuous parts of the transition probabilities $\{ P(x, \cdot ) \}_{x\in X}$ are continuous with respect to $x$. Under some concentration assumptions on the asymptotic transition probabilities $P^m(y , \cdot ) $, which also satisfy $\inf (\mathop{\rm supp}\nolimits Pf_1 ) \preceq \inf (\mathop{\rm supp}\nolimits Pf_2 )$ whenever $ \inf (\mathop{\rm supp}\nolimits f_1)\preceq \inf (\mathop{\rm supp}\nolimits f_2) $, we prove that the iterates $P^n$ converge in the weak$^{\ast }$ operator topology.
