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Some zero-one laws for additive functionals of Markov processes
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  • Published: December 1994

Some zero-one laws for additive functionals of Markov processes

  • Rainer Höhnle1 &
  • Karl-Theodor Sturm2 

Probability Theory and Related Fields volume 100, pages 407–416 (1994)Cite this article

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Summary

Let (X x t,P ) be anm-symmetric Markov process with a strictly positive transition density. Consider the additive functionalA t : = ∫ t0 f (X s ) wheref:E→[0, ∞] is a universally measurable function on the state spaceE. Among others, we prove thatP x(A t <∞)=1, for somex∈E and somet>0, already impliesP x(A t <∞)=1, for quasi everyx∈E and allt>0. The latter is also equivalent toP x(A t <∞)>0, for quasi everyx∈E and allt>0, and to the analytic condition\(\smallint _{F_n } fdm< \infty \), for a sequence of finely open Borel setsF n such thatE∪F n is polar. In the special cases of Brownian motion and Bessel process, these results were obtained earlier by H.J. Engelbert, W. Schmidt, X.-X. Xue and the authors.

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Author information

Authors and Affiliations

  1. Mathematisch-Geographische Fakultät, Universität Eichstätt, D-85071, Eichstätt, Germany

    Rainer Höhnle

  2. Mathematisches Institut, Universität Erlangen-Nürnberg, Bismarckstrasse 1 1/2, D-91054, Erlangen, Germany

    Karl-Theodor Sturm

Authors
  1. Rainer Höhnle
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  2. Karl-Theodor Sturm
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Höhnle, R., Sturm, KT. Some zero-one laws for additive functionals of Markov processes. Probab. Th. Rel. Fields 100, 407–416 (1994). https://doi.org/10.1007/BF01268986

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  • Received: 21 February 1994

  • Revised: 21 July 1994

  • Issue Date: December 1994

  • DOI: https://doi.org/10.1007/BF01268986

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Mathematics Subject Classification

  • 60 J 55
  • 60 J 57
  • 60 J 40
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