Finely \(\mu\)-harmonic functions of a Markov process (Q2759047)

In classical potential theory, the harmonic functions are defined using the Laplacian, and may be characterized by a mean value property or by a Poisson-type integral representation. The purpose of the paper under review is to provide similar results in the framework of a transient, Borel right process with state space \(E\); \(m\) denotes a fixed \(\sigma \)-finite excessive measure; \(G\) is a finely open, nearly Borel set and \(\mu \) is a locally smooth measure on \(E\). Then a quasi-finely continuous function \(h\) on \(E\) is named \(\mu \)-finely harmonic function on \(G\) if \((\Lambda _{G} + \mu) h = 0\) where \(\Lambda _{G}\) is the extended generator of the process, restricted to \(G\) [as defined by the author, Electron. J. Probab. 4, No. 19 (1999; Zbl 0936.60066)]. A first characterization says that \(h\) is a \(\mu \)-finely harmonic function on \(G\) iff, up to an \(m\)-polar set, \(G\) is a countable union of sets \(G_{n}\) such that: NEWLINE\[NEWLINEh = P_{\tau _{n}} h + E^{\bullet } \int _{0}^{\tau _{n}} h(X_{t}) dA_{t}^{n}NEWLINE\]NEWLINE on \(G_{n}\) (where \(A^{n}\) is the CAF corresponding to \(1_{G_{n}} \mu \); \(\tau _{n}\) is the exit time from \(G_{n}\) and \(P_{\tau _{n}} h = E^{\bullet}[ h(X_{\tau _{n}}) ]\)). NEWLINENEWLINENEWLINEA separate section is devoted to the study of the case when the \(m\)-polar set from the above result may be taken empty. A second characterization says, under some mild technical assumptions, that \(h\) is a \(\mu \)-finely harmonic function on \(G\) iff \(h = E^{\bullet}[ \exp (A_{\tau _{n}} ^{n})h(X_{\tau _{n}})]\) on \(G_{n}\). Both representations reduce to \(h = P_{\tau _{n}} h\) on \(G_{n}\), when \(\mu = 0\). Moreover, \(h:= P_{\tau }^{A} g\) on \(G\) and \(h:=g\) on \(X\setminus G\) solves a Dirichlet type problem with given data \(g\) on \(X\setminus G\).