THE MAXIMAL EXPECTED LIFETIME OF BROWNIAN MOTION

Abstract

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