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On the Deterministic and Asymptotic σ-Algebras of a Markov Operator

Published online by Cambridge University Press:  20 November 2018

Ulrich Krengel  and
Michael Lin
Ulrich Krengel
Affiliation:
Institut für Mathematische Stochastik, Lotzestrasse 13, D-3400 Göttingen, West Germany
Michael Lin
Affiliation:
Dept. of Mathematics and Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
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Abstract

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Let P be a Markov operator on L∞(X, Σ, m) which does not disappear (i.e., P1A ≡ 0 => 1A ≡ 0 ) . We study the relationship between the σ-algebras

(the deterministicσ-algebra), and the asymptoticσ-algebra

When m is a σ-finite invariant measure, measurable iff p*npnf = f, and also iff Pnf has the same distribution as f . The case of a convolution operator on a locally compact group is considered.

Keywords

60J0560J15:47A35
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 32 , Issue 1 , 01 March 1989 , pp. 64 - 73
Copyright
Copyright © Canadian Mathematical Society 1989

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