A subharmonicity property of harmonic measures (Q2790923)

The author gives an alternative proof of the following result obtained in [\textit{D. Benko} et al., Rev. Mat. Iberoam. 28, No. 4, 947--960 (2012; Zbl 1253.31004)].NEWLINENEWLINETheorem 1. Let \(S\) be a circle in the plane, \(F\subset S\) a closed subset. If \(a\in S\setminus F\) and \(I\subset F\) is an arc, then the density of the harmonic measure \(\omega(\cdot, a,{\mathbb R}^2\setminus F)\) with respect to the arc-measure on \(S\) is convex in \(I\).NEWLINENEWLINEThe approach relies on random walks. Also, the author obtains the following higher-dimensional version of the above result.NEWLINENEWLINETheorem 2. Let \(S\) be the \((n-1)\)-dimensional sphere in \({\mathbb R}^n\), \(n\geq 3\), and \(F\subset S\) a closed subset. If \(a\in S\setminus F\), then the density of the harmonic measure \(\omega(\cdot, a,{\mathbb R}^n\setminus F)\) is subharmonic on the \((n-1)\)-dimensional interior of \(F\).