On iterates of strong Feller operators on ordered phase spaces

Wojciech Bartoszek. "On iterates of strong Feller operators on ordered phase spaces." Colloquium Mathematicae 101.1 (2004): 121-134. <http://eudml.org/doc/286258>.

@article{WojciechBartoszek2004,
abstract = {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 $\{P(x,·)\}_\{x∈X\}$ are continuous with respect to x. Under some concentration assumptions on the asymptotic transition probabilities $P^\{m\}(y,·)$, 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.},
author = {Wojciech Bartoszek},
journal = {Colloquium Mathematicae},
keywords = {Markov operator; asymptotic stability; 0-2 law; Foguel alternative; cell cycle},
language = {eng},
number = {1},
pages = {121-134},
title = {On iterates of strong Feller operators on ordered phase spaces},
url = {http://eudml.org/doc/286258},
volume = {101},
year = {2004},
}

TY - JOUR
AU - Wojciech Bartoszek
TI - On iterates of strong Feller operators on ordered phase spaces
JO - Colloquium Mathematicae
PY - 2004
VL - 101
IS - 1
SP - 121
EP - 134
AB - 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 ${P(x,·)}_{x∈X}$ are continuous with respect to x. Under some concentration assumptions on the asymptotic transition probabilities $P^{m}(y,·)$, 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.
LA - eng
KW - Markov operator; asymptotic stability; 0-2 law; Foguel alternative; cell cycle
UR - http://eudml.org/doc/286258
ER -