Finely harmonic morphisms, Brownian path preserving functions and conformal martingales (Q792709)

If U denotes a finely open set in \({\mathbb{C}}\), call a finely continuous function \(\phi\) :\(U\to {\mathbb{C}}\) a finely harmonic morphism if, for any finely harmonic function h on a finely open set W, the function h\({\mathbb{O}}\phi\) is finely harmonic in \(\phi^{-1}(W)\). Let \(\tau_ U\) denote the first exit time of Brownian motion from U. The author proves the following expected stochastic characterizations of finely harmonic morphisms: \(\alpha)\) Finely harmonic morphisms are the only functions which preserve the paths of the Brownian motion until \(\tau_ U\). \(\beta)\) Finely harmonic morphisms are the only functions \(\phi\) such that \(Z_ t=\phi(B_{t\wedge \tau_ U})\) defines a conformal martingale, i.e. a martingale such that \(<Z,\bar Z>=0.\) A conjecture is formulated in the frame of the fine topology of an harmonic space with adapted Markov process. As an application of his theorem, the author gives a striking result in the frame of asymptotic values, which, even restricted to the very special case of an analytic function in \({\mathbb{C}}^ n\), appears to be a new one.