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 (τ 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 (τ 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 B t which generalizes the notion of a cocapacitary measure such that\(\hat C\)(B)=\(\mathop {\sup }\limits_{t > 0} \) ρ B t (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.
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Research supported in part by NSF Grant DMS 8419377
Research carried out while visiting University of California, San Diego, during Spring 1985
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Getoor, R.K., Steffens, J. Capacity theory without duality. Probab. Th. Rel. Fields 73, 415–445 (1986). https://doi.org/10.1007/BF00776241
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DOI: https://doi.org/10.1007/BF00776241
Keywords
- State spaceE
- Stochastic Process
- Stationary Process
- Probability Theory
- Mathematical Biology