Invariant densities and mean ergodicity of Markov operators

Emel'yanov E.

ISRAEL JOURNAL OF MATHEMATICS, vol.136, pp.373-379, 2003 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 136
  • Publication Date: 2003
  • Doi Number: 10.1007/bf02807206
  • Journal Name: ISRAEL JOURNAL OF MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.373-379
  • Middle East Technical University Affiliated: No

Abstract

We prove that a, Markov operator T on L-1 has an invariant density if and only if there exists a density f that satisfies lim sup(n-->infinity) parallel toT(n) f-fparallel to < 2. Using this result, we show that a Frobenius-Perron operator P is mean ergodic if and only if there exists a density w such that lim sup(n-->infinity) parallel toP(n)f - wparallel to < 2 for every density f. Corresponding results hold for strongly continuous semigroups.