Capacity theory without duality
type: Item
instance of: scholarly article
title: Capacity theory without duality
zbMATH Open document ID: 0586.60069
DOI: 10.1007/BF00776241
author: J. Steffens
author: R. K. Getoor
publication date: 1986-01-01T00:00:00Z
review text: 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 relative to an excessive measure. If \((P_ t)_{t\geq o}\) denotes the semigroup of the process on the state space E and if m is an excessive measure on E, then there exists a process \(Y=(Y_ t)_{t\in {\mathbb{R}}}\) on E with random birth and death and a \(\sigma\)-finite measure \(Q_ m\) such that Y is stationary under \(Q_ m\) and Markov with respect to \((P_ t).\) For a set B in E the hitting (resp. last exit) time of Y is denoted by \(\tau_ B\) (resp. \(\lambda_ B)\), and B is called transient (resp. cotransient) if \(Q_ m(\lambda_ B=\infty)=0\) (resp. \(Q_ m(\tau_ B=-\infty)=0)\). The main theorem then states that for a both transient and cotransient set B the distributions of \(\lambda_ B\) and \(\tau_ B\) under \(Q_ m\) are the same. For such B the capacity is defined by \(C(B):=Q_ m(\lambda_ B\in [0,1])\) and the cocapacity by \(\hat C(B):=Q_ m(\tau_ B\in [0,1]),\) 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 \((\rho^ B_ t)\) which generalizes the notion of a cocapacitary measure such that \(\hat C(\)B)\(=\sup_{t\downarrow 0}\rho^ B_ t(1).\) The paper contains investigations of transience and cotransience, a decomposition of the cocapacitary entrance law, some remarks on left versions, and furthermore a generalization of Spitzer's asymptotic formula.
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 3940354
zbMATH Keywords: theory of capacity for a Borel right process
zbMATH Keywords: excessive measure
zbMATH Keywords: cocapacitary entrance law
zbMATH Keywords: cocapacitary measure
zbMATH Keywords: transience and cotransience
zbMATH Keywords: generalization of Spitzer's asymptotic formula
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 3940354