Brownian motion and the fundamental frequency of a drum (Q1341281)

Let \(\lambda_ D\) be the first Dirichlet eigenvalue for the Laplacian in the simply connected domain \(D\). Let \(\tau_ D\) be the first exit time of the Brownian motion from \(D\) and \(E_ z(\tau_ D)\) be the expectation of \(\tau_ D\) under the measure of this Brownian motion starting at the point \(z\) in \(D\). Some estimates for the eigenvalue \(\lambda_ D\) and for the expectation \(E_ z(\tau_ D)\) are established.