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Maximizing mean exit-time of the Brownian motion on Riemannian manifolds. (English) Zbl 1317.49053

For an \(n\)-dimensional Riemannian manifold \((M,g)\) let \(dv_g\) be the associated Riemannian measure and let \(\Omega\) be any compact connected domain in \(M\) with smooth boundary \(\partial\Omega\). Let \(C^\infty_c(\Omega)\) be the space of \(C^\infty\)-functions with compact support and \(H^2_{1,c}(\Omega)\) its completion with respect to the Sobolev norm. If \(f_\Omega\) is a solution of the Dirichlet problem \(\Delta f_{|\Omega}=1\) and \(f_{|\partial\Omega}=0\), where \(\Delta\) is the Laplacian on \(M\), then \(f_\Omega\in H^2_{1,c}(\Omega)\) and it is the unique critical point of the functional \(E_\Omega(f)=\frac1{\text{Vol}(\Omega)}\left(\int_\Omega 2(f-|\nabla f|^2)dv_g\right)\). The value \(\mathcal{E}(\Omega)=\max\limits_{f\in H^2_{1,c}(\Omega)}(E_\Omega(f))=E_\Omega(f_\Omega)\) of the functional \(\mathcal{E}\) is called the mean exit-time from \(\Omega\) of the Brownian motion, or the mean exit-time from \(\Omega\), for short. For any complete Riemannian manifold \((M,g)\) and any point \(x\in M\) the set \(\mathbb S_x\) is the unit sphere of the Euclidean space \((T_xM,g_x)\), and for any \(v\in\mathbb S_x\) the maximum of the \(T\)’s such that the geodesic \(c_v:[0,T]\to M\) is minimizing is denoted by \(\text{Cut}(v)\). \((M,g)\) is said to be harmonic at \(x\) if \(v\mapsto\text{Cut}(v)\) is a constant on \(\mathbb S_x\) and if any geodesic sphere centered at \(x\) of radius \(r<\text{Cut}(v)\) is a smooth hypersurface with constant mean curvature. \(M\) is harmonic if it is harmonic at each of its points. \((M,g)\) is said to be isoperimetric at \(x\) if it is harmonic at \(x\) and if, for any compact domain \(\Omega\subset M\) with smooth boundary, the geodesic ball \(\Omega^*\) centered at \(x\) with the same volume as \(\Omega\) satisfies \(\text{Vol}_{n-1}(\partial\Omega^*)\leq\text{Vol}_{n-1}(\partial\Omega)\).
In this paper, the authors study properties of the functional \(\mathcal{E}\) over the set of all compact domains of fixed volume \(v\) in any Riemannian manifold \((M,g)\). They show that if \((M,g)\) is a complete, connected Riemannian manifold whose Ricci curvature satisfies \(\text{Ric}_g\geq(n-1)g\) and for any compact domain \(\Omega\) with smooth boundary in \(M\) a geodesic ball \(\Omega^*\) of the canonical sphere \((\mathbb S^n,g_0)\) such that \(\frac{\text{Vol}(\Omega^*,g_0)}{\text{Vol}(\mathbb S^n,g_0)}=\frac{\text{Vol}(\Omega,g)}{\text{Vol}(M,g)}\), then \(\mathcal{E}(\Omega)\leq\mathcal{E}(\Omega^*)\). Also, they prove that if \((M,g)\) is a compact Riemannian manifold and \(\Omega\) is any compact domain with smooth boundary in \(M\) such that \(\text{Vol}(\Omega)\leq \frac12\text{Vol}(M)\), then \(\mathcal{E}(\Omega)\leq\frac1{H(M,g)^2}\), where \(H(M,g)=\inf\limits_{\Omega}\left(\frac{\text{Vol}_{n-1}(\partial\Omega)}{\min\{\text{Vol}(\Omega),\text{Vol}(M\smallsetminus\Omega)\}}\right)\) is the Cheeger isoperimetric constant.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
60J65 Brownian motion
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References:

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