Additive functionals and entrance laws (Q1063332)

Let \(\nu =\{\nu_ t: t\in {\mathbb{R}}\}\) and \(h=\{h_ t: t\in {\mathbb{R}}\}\) be an entrance and exit rule, respectively, for a stationary Markov transition function in a standard Borel space (E,\({\mathcal E})\). In an appropriate trajectory space, \(\Omega\) :\({\mathbb{R}}\to E\), one may construct a \(\sigma\)-finite measure \({\mathbb{P}}^ h_{\nu}\) such that, roughly speaking, the coordinate process comes into the state space at the random birth time according to the rule \(\nu\), evolves according to the Markovian transition law, and leaves the state space at the random death time according to the rule h. In the paper under review the additive functionals of a process with this structure are studied. The main result gives a constructive one-to-one correspondence between entrance laws and continuous additive functionals. Unfortunately this is a too complicated matter to allow any details within this limited space.