Let (X,d) be a metric space where all closed balls are compact, with a fixed σ-finite Borel measure μ. Assume further that X is endowed with a linear order ⪯. Given a Markov (regular) operator P: L¹(μ) → L¹(μ) we discuss the asymptotic behaviour of the iterates Pⁿ. The paper deals with operators P which are Feller and such that the μ-absolutely continuous parts of the transition probabilities are continuous with respect to x. Under some concentration assumptions on the asymptotic transition probabilities , which also satisfy inf(supp Pf₁) ⪯ inf(supp Pf₂) whenever inf(supp f₁) ⪯ inf(supp f₂), we prove that the iterates Pⁿ converge in the weak* operator topology.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm101-1-8,
author = {Wojciech Bartoszek},
title = {On iterates of strong Feller operators on ordered phase spaces},
journal = {Colloquium Mathematicae},
volume = {100},
year = {2004},
pages = {121-134},
zbl = {1057.37005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm101-1-8}
}
Wojciech Bartoszek. On iterates of strong Feller operators on ordered phase spaces. Colloquium Mathematicae, Tome 100 (2004) pp. 121-134. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm101-1-8/