Killed Brownian motion and the Brunn-Minkowski inequalities (Q2898406)

The purpose here is to establish Brunn-Minkowski inequalities by means of killed Brownian motions.NEWLINENEWLINE Let \(K_0\), \(K_1\) be compact sets in \(\mathbb{R}^n\), \(0<\alpha<1\), \(K_\alpha:= (1-\alpha)K_0+ K_1\), and, for \(j\in\{0,\alpha,1\}\), consider continuous functions \(h_j\geq 0\) on \(K_j\) and \(f_j> 0\) on \(\partial K_j\) such that NEWLINE\[NEWLINEh_\alpha((1- \alpha) x_0+\alpha x_1)\leq\min\{h_0(x_0), h_1(x_1)\}\quad\text{if }x_j\in K_jNEWLINE\]NEWLINE and NEWLINE\[NEWLINEf_\alpha((1- \alpha)x_0+ \alpha x_1)\geq (1-\alpha) f_0(x_0)+ \alpha f_1(x_1)\quad\text{if }x_j\in\partial K_j.NEWLINE\]NEWLINE The main result asserts that, if (for \(j\in\{0,\alpha,1\}\)) NEWLINE\[NEWLINE\Delta\psi_j= h_j\psi_j\quad\text{on }\text{int}(K_j)\quad\text{and}\quad \psi_j= f_j\quad\text{on }\partial K_j,NEWLINE\]NEWLINE then \(\psi_\alpha((1-\alpha) x_0+\alpha x_1)\geq \psi_0(x_0)^{1-\alpha}\psi_1(x_1)^\alpha\) (for any \(x_j\in K_j\)).NEWLINENEWLINE Variants of this theorem are also given, namely in the cases when \(\Delta\psi_j= g_j\leq 0\) on \(\text{int}(K_j)\), or when \(\Delta\psi_j= h_j\psi_j+ g_j\) and \(f_j\equiv 0\).