Jensen Measures in Potential Theory

Abstract

It is shown that, for open sets in classical potential theory and—more generally—for elliptic harmonic spaces Y, the set J x (Y) of Jensen measures (representing measures with respect to superharmonic functions on Y) for a point x ∈ Y is a simple union of closed faces of the compact convex set $M_x(\mathcal P(Y))$ of representing measures with respect to potentials on Y, a set which has been thoroughly studied a long time ago. In particular, the set of extreme Jensen measures can be immediately identified. The results hold even without ellipticity (thus capturing also many examples for the heat equation) provided a rather weak approximation property for superharmonic functions holds. Equally sufficient are a certain transience property and a weak regularity property. More important, each of these properties turns out to be necessary and sufficient for obtaining (in the classical case) that J x (Y) coincides with the set of all compactly supported probability measures in $M_x(\mathcal P(Y))$ .