On iterates of strong Feller operators on ordered phase spaces

Tom 101 / 2004

Wojciech Bartoszek Colloquium Mathematicum 101 (2004), 121-134 MSC: 37A30, 47A35, 62P10, 60J35. DOI: 10.4064/cm101-1-8


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.


  • Wojciech BartoszekDepartment of Mathematics
    Gdańsk University of Technology
    Narutowicza 11/12
    80-952 Gdańsk, Poland

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