Maximizing mean exit-time of the Brownian motion on Riemannian manifolds

Abstract

We study the functional \(\Omega \mapsto {\mathcal E} (\Omega )\), where \(\Omega \) runs in the set of all compact domains of fixed volume \(v\) in any Riemannian manifold \((M,g)\) and where \({\mathcal E} (\Omega )\) is the mean exit-time of the Brownian motion (also called torsional rigidity) of \(\Omega \). We first prove that, when \((M,g)\) is strictly isoperimetric at one of its points, the maximum of this functional is realized by the geodesic ball centered at this point. When \((M,g)\) is any Riemannian manifold, for every domain \(\Omega \) in \(M\), we prove that \({\mathcal E} (\Omega ) \le {\mathcal E} (\Omega ^*)\), where \(\Omega ^*\) is the corresponding symmetrized domain on a model-space \((M^*,g^*)\). We also consider the functional \(\Omega \mapsto {\mathcal E} (\Omega )\), when \(\Omega \) runs in the set of all compact domains, with smooth boundary in the class of all Riemannian manifolds with “bounded” geometry. We prove two results in this direction. In the first one (Theorem 1.9) we prove that for every complete, connected Riemannian manifold \((M,g)\) whose Ricci curvature satisfies \(\mathrm{Ric}_g \ge (n-1)g\) and for every compact domain with smooth boundary \(\Omega \) in \(M\) one has \({\mathcal E} (\Omega ) \le {\mathcal E} (\Omega ^*)\), where \(\Omega ^*\) is a geodesic ball of the canonical sphere \((\mathbb S^n , g_0 )\) such that \(\dfrac{\mathrm{Vol} (\Omega ^*, g_0)}{\mathrm{Vol} (\mathbb S^n , g_0 )} = \dfrac{\mathrm{Vol} (\Omega , g )}{\mathrm{Vol} (M , g )}\). Morever, if there exists some domain \(\Omega \subset M\) such that \({\mathcal E} (\Omega ) = {\mathcal E} (\Omega ^*)\) then \( (M, g) \) is isometric to \((\mathbb S^n , g_0 )\) and \(\Omega \) is isometric to \(\Omega ^*\). The second result (Theorem 1.10) shows that if \((M,g)\) is any compact Riemannian manifold and \(\Omega \) is any compact domain with smooth boundary in \(M\) such that \(\mathrm{\ Vol} (\Omega ) \le \frac{1}{2} \mathrm{\ Vol} (M)\), then \({\mathcal E} (\Omega ) \le \dfrac{1}{H(M,g)^2}, \) where \(H(M,g)\) is Cheeger’s isoperimetric constant.