Lifetime asymptotics of iterated Brownian motion in

¹ Department of Mathematics, Purdue University, West Lafayette, IN 47906, USA; enane@math.purdue.edu
² Current address: Department of Statistics and Probability, Michigan State University, A413 Wells Hall, East Lansing, MI 48824-1027, USA; nane@stt.msu.edu

Received: 5 May 2006
Revised: 8 September 2006

Abstract

Let be the first exit time of iterated Brownian motion from a domain started at and let be its distribution. In this paper we establish the exact asymptotics of over bounded domains as an improvement of the results in DeBlassie (2004) [DeBlassie, Ann. Appl. Prob. 14 (2004) 1529–1558] and Nane (2006) [Nane, Stochastic Processes Appl. 116 (2006) 905–916], for 
 
where . Here λ_D is the first eigenvalue of the Dirichlet Laplacian in D, and ψ is the eigenfunction corresponding to λ_D. We also study lifetime asymptotics of Brownian-time Brownian motion, , where X_t and Y_t are independent one-dimensional Brownian motions, in several unbounded domains. Using these results we obtain partial results for lifetime asymptotics of iterated Brownian motion in these unbounded domains.