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On the equivalence of three potential principles for right Markov processes
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  • Published: June 1990

On the equivalence of three potential principles for right Markov processes

  • P. J. Fitzsimmons1 

Probability Theory and Related Fields volume 84, pages 251–265 (1990)Cite this article

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Summary

We examine three of the principles of probabilistic potential theory in a nonclassical setting. These are: (i) the bounded maximum principle, (ii) the positive definiteness of the energy (of measures of bounded potential), and (iii) the condition that each semipolar set is polar. These principles are known to be equivalent in the context of two Markov processes in strong duality, when excessive functions are lower semicontinuous. We show that when the principles are appropriately formulated their equivalence persists in the wider context of a Borel right Markov processX with distinguished excessive measurem. We make no duality hypotheses andm need not be a reference measure. Our main tools are the stationary process (Y, Q m) associated withX andm, and a correspondence between potentials μU and certain random measures over (Y, Q m).

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Authors and Affiliations

  1. Department of Mathematics, C-012, University of California, San Diego, 92093, La Jolla, CA, USA

    P. J. Fitzsimmons

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  1. P. J. Fitzsimmons
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Research supported in part by NSF Grant 8419377

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Fitzsimmons, P.J. On the equivalence of three potential principles for right Markov processes. Probab. Th. Rel. Fields 84, 251–265 (1990). https://doi.org/10.1007/BF01197847

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  • Received: 24 February 1988

  • Revised: 07 July 1989

  • Issue Date: June 1990

  • DOI: https://doi.org/10.1007/BF01197847

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Keywords

  • Stochastic Process
  • Maximum Principle
  • Markov Process
  • Mathematical Biology
  • Potential Theory
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