On iterates of strong Feller operators on ordered phase spaces

Wojciech Bartoszek¹

¹ Department of Mathematics Gdańsk University of Technology Narutowicza 11/12 80-952 Gdańsk, Poland

Colloquium Mathematicum, Tome 101 (2004) no. 1, pp. 121-134.

Résumé

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.

Détail

DOI : 10.4064/cm101-1-8

Keywords: metric space where closed balls compact fixed sigma finite borel measure assume further endowed linear order nbsp preceq given markov regular operator discuss asymptotic behaviour iterates paper deals operators which feller absolutely continuous parts transition probabilities cdot continuous respect nbsp under concentration assumptions asymptotic transition probabilities cdot which satisfy inf mathop supp nolimits preceq inf mathop supp nolimits whenever inf mathop supp nolimits preceq inf mathop supp nolimits prove iterates converge weak ast operator topology