Capacity theory without duality
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.
theory of capacity for a Borel right process
generalization of Spitzer's asymptotic formula
scientific article; zbMATH DE number 3940354
scientific article