Maximizing mean exit-time of the Brownian motion on Riemannian manifolds
Maximizing mean exit-time of the Brownian motion on Riemannian manifolds
复制标题
黎曼流形上布朗运动的平均退出时间最大化
DOI：
10.1007/s00605-014-0722-3
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
A. Loi
中科院分区：
文献类型：
--
作者：
L. Cadeddu;S. Gallot;A. Loi
本文研究了在任意黎曼流形上的所有定体积紧域的集合中的泛函，其中是的布朗运动（也称为扭刚性）的平均出口时。我们首先证明了，当在其中一点上是严格等周的时，该泛函的极大值是由以该点为中心的测地线球实现的。当有任意黎曼流形时，对每一个Domain，我们证明了在模型空间上有相应的对称化Domain。我们还考虑了在所有紧区域的集合中运行的具有光滑边界的泛函在所有具有“有界”几何的黎曼流形类中的情形。我们在这个方向上证明了两个结果。在第一部分（定理1.9）中，我们证明了：对每一个Ricci曲率满足的完备连通黎曼流形和每一个具有光滑边界的紧致区域，其中有一个典范球面的测地球，使得。此外，如果存在某个域，使得与和等距。第二个结果（定理1.10）表明，如果是任一紧致黎曼流形，且是任一具有光滑边界的紧致区域，使得，则其中是Cheeger等周常数。
We study the functional, whereruns in the set of all compact domains of fixed volumein any Riemannian manifoldand whereis themean exit-time of the Brownian motion(also calledtorsional rigidity) of. We first prove that, whenis strictly isoperimetric at one of its points, the maximum of this functional is realized by the geodesic ball centered at this point. WhenisanyRiemannian manifold, for every domainin, we prove that, whereis the corresponding symmetrized domain on a model-space. We also consider the functional, whenruns 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 manifoldwhose Ricci curvature satisfiesand for every compact domain with smooth boundaryinone has, whereis a geodesic ball of the canonical spheresuch that. Morever, if there exists some domainsuch thatthenis isometric toandis isometric to. The second result (Theorem 1.10) shows that ifis any compact Riemannian manifold andis any compact domain with smooth boundary insuch that, thenwhereis Cheeger’s isoperimetric constant.