Harmonic functions and exit boundary of superdiffusion

All positive harmonic functions in an arbitrary domain of a Euclidean space can be described in terms of the so-called exit boundary. This was established in 1941 by R.S. Martin. A probabilistic approach to the Martin theory is due to Doob and Hunt. It was extended later to harmonic functions associated with a wide class of Markov processes. The subject of this paper are harmonic functions associated with a superdiffusion X (we call them X-harmonic). The results of Evans and Perkins imply the existence and uniqueness of an integral representation of positive X-harmonic functions through extreme functions. An outstanding problem is to find all extremal functions (they are in 1–1 correspondence with the points of the exit boundary). An interest in X-harmonic functions is motivated, in part, by the fact that each of them provides a way of conditioning a superprocess. Path properties of conditioned superprocesses (corresponding to various special X-harmonic functions) were investigated by a number of authors. Important classes of X-harmonic functions are related to positive solutions of semilinear partial differential equations. Almost nothing is known about their decomposition into extreme elements. A progress in this direction may create new tools for investigating solutions of the equations. The goal of this paper is to summarize all known facts about X-harmonic functions, to present the results of various authors in a more general setting in a unified form and to outline a program of further work.