A stochastic characterization of harmonic morphisms (Q1107895)

It is proved that the harmonic morphisms between two \({\mathcal P}\)-harmonic spaces (\({\mathfrak X},{\mathfrak U})\) and (\({\mathfrak Y},{\mathfrak V})\) are exactly the continuous functions \(\phi: {\mathfrak X}\to {\mathfrak Y}\) which map an extended Ray process \(Z_ t\) whose excessive functions are \({\mathfrak U}\)- hyperharmonic into the diffusion \(Y_ t\) associated to \(({\mathfrak Y},{\mathfrak V})\). The branch set B of \(Z_ t\) is identified as the set of points where \(\phi\) is finely locally constant. We also obtain that for any \(x\in {\mathfrak X}\) either \(\phi\) is finely locally constant at x or \(\phi\) maps every fine neighbourhood of x onto a fine neighbourhood of \(\phi(x)\). If \(B=\emptyset\), i.e. \(\phi\) is finely locally non-constant, then \(Z_ t\) can be replaced by a continuous time change of the diffusion \(X_ t\) associated to \(({\mathfrak X},{\mathfrak U})\). This allows one to use stochastic methods in the investigation of harmonic morphisms. For example, as an application we prove that if H is a polar set in \({\mathfrak Y}\), then \(\phi^{-1}(H)\setminus B\) is a polar set in \({\mathfrak X}\). We also obtain a general boundary value result for harmonic morphisms.