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A Decomposition of Excessive Measures

  • R. M. Blumenthal
Part of the Progress in Probability and Statistics book series (PRPR, volume 12)

Abstract

Let {Pt;t ≥ 0} denote the transition semigroup for a Borel right Markov process on a state space (E, ℰ). We set \( U\left( {x,D} \right) = \int\limits_0^{\infty } {{P_t}} \left( {x,D} \right)dt \) and denote by µU and Uf the potential of a measure μ and a function f respectively. Given a measure m on ℰ and a set D in ℰ let mD be the measure mD(B) = m(D∩B). A measure m on ℰ is called excessive if it is σ-finite and m ≥ mPt for all t. If µ is a measure on ℰ the formula \( \mu U(A) = \int {\mu \left( {dx} \right)U\left( {x,A} \right)} \) defines a measure, the potential of µ, on ℰ. It will be excessive if and only if it is σ-finite. An excessive measure m is called invariant if mPt = m for all t.

Keywords

Markov Process Borel Function Finite Measure Countable Union Invariance Result 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Fitzsimmons, P. J. and Maisonneuve, B. “Excessive Measures and Markov Processes with Random Birth and Death”, to appear in Z. Wahrscheinlichkeitstheorie verw. Gebiete.Google Scholar
  2. Getoor, R. K. and Sharpe, M. J. “Naturality, standardness and weak duality for Markov processes.” Z. Wahrscheinlichkeitstheorie verw. Gebiete, 67 (1984), 1–62.MathSciNetMATHCrossRefGoogle Scholar
  3. Getoor, R. K. and Steffens, J. “Capacity Theory without Duality”, to appear.Google Scholar

Copyright information

© Birkhäuser Boston, Inc. 1986

Authors and Affiliations

  • R. M. Blumenthal
    • 1
  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

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