Quantitative Inequalities for the Expected Lifetime of Brownian Motion

Abstract

The isoperimetric inequalities for the expected lifetime of Brownian motion state that the ${\mathit{L}}^{\mathit{p}}$-norms of the expected lifetime in a bounded domain for $1\le \mathit{p}\le \infty$ are maximized when the region is a ball with the same volume. In this paper, we prove quantitative improvements of the inequalities. Since the isoperimetric properties hold for a wide class of Lévy processes, many questions arise from these improvements.