THE MAXIMAL EXPECTED LIFETIME OF BROWNIAN MOTION
THE MAXIMAL EXPECTED LIFETIME OF BROWNIAN MOTION
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布朗运动的最大预期寿命
DOI：
10.3318/pria.2011.111.1
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
T. Carroll
中科院分区：
文献类型：
--
作者：
R. Bañuelos;T. Carroll
它表明，有最大化的布朗运动的期望寿命在所有单连通域的给定的内径和所有的起点的域。1.背景和声明的主要结果布朗运动的预期退出时间从一个简单的连接平面域不能大，除非域包含一个大的磁盘。这个非常直观的性质有许多应用，特别是拉普拉斯算子的特征值。布朗运动从区域D的退出时间将被表示为td，并且对于具有初始点z的路径，关于维纳测度的期望将被表示为E2。我们将D中包含的所有圆盘的上确界半径记为Rd。这与Rd = sup {fo（z）：2 e DJ相同，其中Sd（z）表示从D中的点z到D的边界的距离。这个几何量称为D的半径。从[6]中的论证可以得出，存在一个常数C，使得sup Eztq <CR2d（1.1）zen '相应作者，电子邮件：t.卡罗尔@www.example.com doi：10.3318/PRIA.2011.111.1.引用如下：Rodrigo Banuelos and Tom Carroll，The maximum expected lifetime of Brownian motion，Mathematical Proceedings of the Royal Irish Academy 1-11A（2011），1 - 11; doi：10.3318/PRIA.2011.111.1 Mathematical Proceedings of the Royal Irish Academy，1-11A（1），1 - 11（2011）© Royal Irish Academy此内容于2016年10月19日星期三04：32：53 UTC从www.example.com下载所有使用受限于www.example.com 2 Mathematical Proceedings of the Royal Irish Academy for all simple connected domain D in the plane.特别地，在[5]中示出了该不等式在C = 8.62时成立。文[3]证明了sup E，r/j <（3.228）B~2D。(1.2)Zed
It is shown that there are domains that maximise the expected lifetime of Brownian motion over all simply connected domains of given inradius and all starting points. 1. Background, and statement of the main result The expected exit time of Brownian motion from a simply connected planar domain cannot be large unless the domain contains a large disk. This very intuitive property has had many applications, particularly to eigenvalues of the Laplacian. The exit time of Brownian motion from a domain D will be denoted by td and the expectation with respect to Wiener measure for paths with initial point z will be denoted by E2. We will write Rd for the supremum radius of all disks contained in D. This is the same as Rd = sup{fo(z) : 2 e DJ, where Sd(z) denotes the distance from the point z in D to the boundary of D. This geometric quantity is called the iiiradius of D. It follows from the argument in [6] that there exists a constant C such that sup Eztq < CR2d (1.1) zen 'Corresponding author, e-mail: t.carroll@ucc.ie doi: 10.3318/PRIA.2011.111.1. Cite as follows: Rodrigo Banuelos and Tom Carroll, The maximal expected lifetime of Brownian motion, Mathematical Proceedings of the Royal Irish Academy 111A (2011), 1-11; doi: 10.3318/PRIA.2011.111.1 Mathematical Proceedings of the Royal Irish Academy, 111A (1), 1-11 (2011) © Royal Irish Academy This content downloaded from 207.46.13.184 on Wed, 19 Oct 2016 04:32:53 UTC All use subject to http://about.jstor.org/terms 2 Mathematical Proceedings of the Royal Irish Academy for all simply connected domains D in the plane. In particular, it is shown in [5] that this inequality holds with C = 8.62. In [3] it is shown that sup E,r/j < (3.228)B2D. (1.2) zeD