Jensen measures in potential theory

TL;DR

This paper characterizes Jensen measures in potential theory, showing they form a union of faces of a convex set, with implications for identifying extreme measures across various elliptic and non-elliptic contexts.

Contribution

It provides a unified geometric description of Jensen measures applicable to classical potential theory and elliptic harmonic spaces, extending to heat equation examples.

Findings

Jensen measures form a union of faces of a convex set

Extreme Jensen measures can be explicitly identified

Results apply even without ellipticity under certain conditions

Abstract

It is shown that, for open sets in classical potential theory and - more generally - for elliptic harmonic spaces, the set of Jensen measures for a point is a simple union of closed faces of a compact convex 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 or a certain transience property holds.