Jensen measures and harmonic measures (Q2757786)

Let \(\Omega\) be an open set in \(\mathbb{R}^d\) \((d\geq 2)\) and let \(x\in \Omega\). A Borel probability measure \(\mu\) with compact support in \(\Omega\) is called a Jensen measure for \(x\), written \(\mu\in J_x(\Omega)\), if \(u(x)\leq \int u d\mu\) for all subharmonic functions \(u\) on \(\Omega\). Also, \(H_x(\Omega)\) denotes the subclass of \(J_x(\Omega)\) consisting of harmonic measures for \(x\) and domains \(D\) which are compactly contained in \(\Omega\). The main result of this nicely written paper is that, if \(\varphi:\Omega\to [-\infty,\infty)\) is universally measurable and locally bounded above, then the infimum of \(\{\int \varphi d\mu: \mu\in J_x(\Omega)\}\) coincides with the infimum of \(\{\int \varphi d\omega: \omega\in H_x(\Omega)\cup \{\delta_x\}\}\). This allows the authors to re-express some of their recent work concerning subharmonic functions and Jensen measures [J. Funct. Anal. 147, 420-422 (1997; Zbl 0873.31007)] in terms of the smaller and more familiar class of harmonic measures. NEWLINENEWLINENEWLINEThe authors go on to study the structure of Jensen measures using Choquet theory. They show that \(H_x(\Omega)\cup \{\delta_x\}\) is strictly contained in \(\text{ext} (J_x(\Omega))\), and that the latter set is strictly contained in the closure of \(H_x(\Omega)\) with respect to the weak*-topology on \(J_x(\Omega)\subset C(\Omega)^*\). They pose the problem of identifying the closure of \(H_x(\Omega)\) in the weak*-topology on \(C(\Omega)^*\) and ask, in particular, whether it coincides with \(J_x(\Omega)\). The paper concludes with a consideration of plurisubharmonic analogues of some of this material.