Comparison theorems for exit times
Comparison theorems for exit times
复制标题
退出时间的比较定理
DOI：
10.1007/pl00001681
复制
发表时间：
2001
期刊：
影响因子：
--
通讯作者：
Michael Schmuckenschläger
中科院分区：
文献类型：
--
作者：
Almut Burchard;Michael Schmuckenschläger
抽象的。研究了布朗运动从集合的大小和形状出发的退出时的界，以及这些界与等周不等式的关系。第一个结果是常曲率球面或双曲空间的子集的出射时间的分布函数的上界是关于相同体积圆盘的出射时间的。这相当于Dirichlet热核的重排不等式。为了将这一不等式与经典的等周不等式联系起来，我们导出了一个用边界热流表示的集合周长的公式。一个辅助结果将Riesz重排不等式推广到重积分。
Abstract. We study bounds on the exit time of Brownian motion from a set in terms of its size and shape, and the relation of such bounds with isoperimetric inequalities. The first result is an upper bound for the distribution function of the exit time from a subset of a sphere or hyperbolic space of constant curvature in terms of the exit time from a disc of the same volume. This amounts to a rearrangement inequality for the Dirichlet heat kernel. To connect this inequality with the classical isoperimetric inequality, we derive a formula for the perimeter of a set in terms of the heat flow over the boundary. An auxiliary result generalizes Riesz' rearrangement inequality to multiple integrals.