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Probability Theory and Related Fields

, Volume 73, Issue 3, pp 415–445 | Cite as

Capacity theory without duality

  • R. K. Getoor
  • J. Steffens
Article

Summary

The paper develops a theory of capacity for a Borel right process without duality assumptions. The basic tool in this approach is a stationary process ralative to an excessive measure.

IfP t )t≧0 denotes the semigroup of the process on the state spaceE and ifm is an excessive measure onE, then there exists a processY = (Y t )t∈ℝ onE with random birth and death and a δ-finite measureQ m such thatY is stationary underQ m and Markov with respect to (P t ).

For a setB inE the hitting (resp. last exit) time ofY is denoted by τ B (resp.λ B ), andB is called transient (resp. cotransient) ifQ m (λ B =∞)= 0 (resp.Q m B = − ∞)=0. The main theorem then states that for a both transient and contransient setB the distributions ofλ B and τ B underQ m are the same. For suchB the capacity is denfined byC(B):=Q m (λ B ∈[0, 1] and the cocapacity by\(\hat C\)(B):=Q m B ∈[0, 1], and it is shown that these definitions in fact generalize previous definitions under duality assumptions.

Without duality assumption there is no representation of the capacitary potential in terms of a capacitary measure, but there exists a cocapacitary entrance law ρ t B which generalizes the notion of a cocapacitary measure such that\(\hat C\)(B)=\(\mathop {\sup }\limits_{t > 0} \) ρ t B (1).

The paper contains investigations of transience and cotransience, a decomposition of the cocapacitrary entrance law, some remarks on left versions, and furthermore a generalization of Spitzer's asymptotic formula.

Keywords

State spaceE Stochastic Process Stationary Process Probability Theory Mathematical Biology 
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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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • R. K. Getoor
    • 1
  • J. Steffens
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaLa JollaUSA
  2. 2.Institut für Statistik u. Dok.Universität DüsseldorfDüsseldorf 1Germany

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