The maximal expected lifetime of Brownian motion (Q2837094)

Let \(\tau_{D}\) denote the exit time of Brownian motion from a domain \(D\), and the expectation with respect to the Wiener measure for paths starting at \(z\) is denoted by \(\mathbb{E}_{z}\). It is known that there is a constant \(C\) such that, for all simply connected plane domains, NEWLINE\[NEWLINE \sup_{z \in D}\mathbb{E}_{z}\tau_{D} \leqslant C R_{D}^{2}, NEWLINE\]NEWLINE where \(R_{D}\) is the inradius of \(D\), that is, the supremum of radii of all disks contained in \(D\). The main result of this paper is that there is a simply connected planar domain \(D\) with finite inradius \(R_{D}\) and a point \(z \in D\) that maximizes the following ratio NEWLINE\[NEWLINE \frac{\sup_{z \in D}\mathbb{E}_{z}\tau_{D}}{R_{D}^{2}}. NEWLINE\]NEWLINE The authors adapt \textit{R. M. Robinson}'s argument in [Duke Math. J. 2, 453--459 (1936; JFM 62.0385.03)] dealing with the planar domains which are images under univalent maps of the unit planar disk. Namely, in this case the inradius satisfies a lower bound involving the Bloch-Landau constant, and this inequality is sharp for so-called extremal domains.