Exit times, moment problems and comparison theorems
Consider a Riemannian manifold \(M\) with volume measure \(dx\), a domain \(D\subset M\) which has a compact closure and a smooth boundary, the associated Brownian exit time \(r_D\), and \[ Q_D(t):= \int_D \mathbb{P}_x(r_D> t)\,dx. \] It is known that \(Q_D(t)= \sum_{\lambda\in Sp(D)} A^e_\lambda e^{-\lambda t/2}\), where \(Sp(D)\) denotes the Dirichlet Laplacian spectrum of \(D\). The author makes the easy observation that \[ \int_D\mathbb{E}_x(r^n_D)\,dx= n!\;\sum_{\lambda\in Sp(D)} A^2_\lambda\times (2/\lambda)^n, \] for all \(n\in\mathbb{N}\), and as a consequence, that a comparison between the sequences \(\int_D\mathbb{E}_x(r^n_D)\,dx\) for different domains \(D\) induces a comparison between their first Dirichlet eigenvalues, their volumes and their boundary areas.
scientific article; zbMATH DE number 6162459
scientific article