Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-25T14:09:27.124Z Has data issue: false hasContentIssue false
  • English
  • Français

Mean ergodic theorems on norming dual pairs

Published online by Cambridge University Press:  01 February 2013

MORITZ GERLACH  and
MARKUS KUNZE
MORITZ GERLACH
Affiliation:
University of Ulm, Institute of Applied Analysis, 89069 Ulm, Germany email moritz.gerlach@uni-ulm.demarkus.kunze@uni-ulm.de
MARKUS KUNZE
Affiliation:
University of Ulm, Institute of Applied Analysis, 89069 Ulm, Germany email moritz.gerlach@uni-ulm.demarkus.kunze@uni-ulm.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We extend the classical mean ergodic theorem to the setting of norming dual pairs. It turns out that, in general, not all equivalences from the Banach space setting remain valid in our situation. However, for Markovian semigroups on the norming dual pair $({C}_{b} (E), \mathscr{M} (E))$ all classical equivalences hold true under an additional assumption which is slightly weaker than the e-property.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 4 , August 2014 , pp. 1210 - 1229
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Albanese, A., Bonet, J. and Ricker, W.. Mean ergodic semigroups of operators. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. 106 (2) (2012), 299–319.Google Scholar
Albanese, A. A., Lorenzi, L. and Manco, V.. Mean ergodic theorems for bi-continuous semigroups. Semigroup Forum 82 (1) (2011), 141–171.Google Scholar
Bogachev, V. I.. Measure Theory. Vol. I, II. Springer, Berlin, 2007.Google Scholar
Eberlein, W. F.. Abstract ergodic theorems and weak almost periodic functions. Trans. Amer. Math. Soc. 67 (1949), 217–240.Google Scholar
Ethier, S. N. and Kurtz, T. G.. Markov Processes: Characterization and Convergence (Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics). John Wiley & Sons, New York, 1986.Google Scholar
Fremlin, D. H., Garling, D. J. H. and Haydon, R. G.. Bounded measures on topological spaces. Proc. Lond. Math. Soc. (3) 25 (1972), 115–136.Google Scholar
Jarchow, H.. Locally Convex Spaces (Mathematische Leitfäden). B. G. Teubner, Stuttgart, 1981.Google Scholar
Kelley, J. L.. General Topology (Graduate Texts in Mathematics, 27). Springer, New York, 1975, reprint of the Van Nostrand, Toronto, Ontario, 1955 edition.Google Scholar
Khan, L. A.. The strict topology on a space of vector-valued functions. Proc. Edinb. Math. Soc. (2) 22 (1) (1979), 35–41.Google Scholar
Komorowski, T., Peszat, S. and Szarek, T.. On ergodicity of some Markov processes. Ann. Probab. 38 (4) (2010), 1401–1443.Google Scholar
Köthe, G.. Topological Vector Spaces. I (Die Grundlehren der Mathematischen Wissenschaften, Band 159). Springer, New York, 1969, translated from German by D. J. H. Garling.Google Scholar
Krengel, U.. Ergodic Theorems (de Gruyter Studies in Mathematics, 6). Walter de Gruyter & Co, Berlin, 1985.Google Scholar
Kunze, M.. Continuity and equicontinuity of semigroups on norming dual pairs. Semigroup Forum 79 (3) (2009), 540–560.Google Scholar
Kunze, M.. A Pettis-type integral and applications to transition semigroups. Czechoslovak Math. J. 61 (2) (2011), 437–459.Google Scholar
Nagel, R. J.. Mittelergodische Halbgruppen linearer Operatoren. Ann. Inst. Fourier (Grenoble) 23 (4) (1973), 75–87.Google Scholar
Revuz, D.. Markov Chains (North-Holland Mathematical Library, 11). North-Holland Publishing Co, Amsterdam, 1975.Google Scholar
Satō, R.. On abstract mean ergodic theorems. Tôhoku Math. J. (2) 30 (4) (1978), 575–581.Google Scholar
Seidler, J.. Ergodic behaviour of stochastic parabolic equations. Czechoslovak Math. J. 47 (2) (1997), 277–316.Google Scholar
Sentilles, F. D.. Bounded continuous functions on a completely regular space. Trans. Amer. Math. Soc. 168 (1972), 311–336.Google Scholar
Szarek, T. and Worm, D. T. H.. Ergodic measures of Markov semigroups with the e-property. Ergod. Th. & Dynam. Sys. 32 (3) (2012), 1117–1135.Google Scholar
Worm, D. T. H. and Hille, S. C.. An ergodic decomposition defined by regular jointly measurable Markov semigroups on Polish spaces. Acta Appl. Math. 116 (1) (2011), 27–53.Google Scholar
Worm, D. T. H. and Hille, S. C.. Ergodic decompositions associated with regular Markov operators on Polish spaces. Ergod. Th. & Dynam. Sys. 31 (2) (2011), 571–597.Google Scholar