Representation of measures by balayage from a regular recurrent point (Q1184103)

Let \((X_ t)\) be a Markov process with state space \(E\), starting from a regular recurrent point 0 (assuming such a point exists). Then there is a canonical invariant measure \(\mu\), and it is assumed that \(\mu\) is a reference measure. Let \(L\) be the local time of 0, suitably normalized. Let \((A_ t)\) be a continuous additive functional with Revuz measure \(\nu\) w.r.t. \(\mu\). Then it is shown that, if \(\nu\) is a subprobability and does not charge 0, \(T=\inf\{t:\;A_ t>L_ t\}\) solves the Skorokhod problem for \(\nu\), i.e. \(X_ T\) has the distribution \(\nu\) on \(\{T<\infty\}\). Interesting variants and examples, and an extension to the transient case, are presented.