Brownian motion in twisted domains

Abstract

The tail behavior of a Brownian motion’s exit time from an unbounded domain depends upon the growth of the “inner radius” of the domain. In this article we quantify this idea by introducing the notion of a twisted domain in the plane. Roughly speaking, such a domain is generated by a planar curve as follows. As a traveler proceeds out along the curve, the boundary curves of the domain are obtained by moving out ± g ( r ) \pm g(r) units along the unit normal to the curve when the traveler is r r units away from the origin. The function g g is called the growth radius. Such domains can be highly nonconvex and asymmetric. We give a detailed account of the case g ( r ) = γ r p g(r) = \gamma r^p , 0 > p ≤ 1 0>p\le 1 . When p = 1 p=1 , a twisted domain can reasonably be interpreted as a “twisted cone.”