Harmonic functions and exit boundary of superdiffusion.
The author considers a superdiffusion \(X\) and its \(X\)-harmonic functions. These functions can be used to condition a superprocess. Some \(X\)-harmonic functions are related to positive solutions of semilinear partial differential equations. Therefore, one would like to describe positive \(X\)-harmonic functions in terms of a so-called exit boundary like in the case of a Martin kernel on the Martin boundary of a Markov process. The representation of positive \(X\)-harmonic functions in terms of extremal functions of the same kind due to Evans and Perkins is a first step in this direction. The paper summarizes known facts of several authors about \(X\)-harmonic functions in a unified form: Dynkin's notion of a convex measurable space; recursive moment formulas for exit measures by Salisbury, Verzani and Dynkin; the Poisson representation of infinitely divisible random measures; absolute continuity results for exit measures; positive solutions to the equation \(Lu=\psi(u)\) due to Salisbury and Verzani; and the only known extremal \(X\)-harmonic function \(H(\mu)=\langle 1,\mu\rangle\) due to Evans. A program for future research is outlined.
scientific article; zbMATH DE number 2057774
scientific article