ESAIM: Probability and Statistics (ESAIM: P&S)

ESAIM: P&S, March 2007, Vol. 11, pp. 147-160
DOI: 10.1051/ps:2007012

Lifetime asymptotics of iterated Brownian motion in $\mathbb$

Erkan Nane^{1, 2}

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 May 5, 2006. Revised September 8, 2006. Published online 31 March 2007.)

Abstract
Let $\tau _{D}(Z)$ be the first exit time of iterated Brownian motion from a domain $D \subset \mathbb$ started at $z\in D$ and let $P_{z}[\tau _{D}(Z) >t]$ be its distribution. In this paper we establish the exact asymptotics of $P_{z}[\tau _{D}(Z) >t]$ 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 $z\in D$

$\displaystyle \lim_{t\to\infty} t^\exp\left(\frac\pi^\lambda_t^\right) P_{z}[\tau_{D}(Z)>t]= C(z),\nonumber$

where $C(z)=(\lambda_)/\sqrt{3 \pi}\left( \psi(z)\int_{D}\psi(y){\rm d}y\right) ^{2}$ . Here $\lambda_{D}$ is the first eigenvalue of the Dirichlet Laplacian $\frac\Delta$ in D, and $\psi$ is the eigenfunction corresponding to $\lambda_{D}$ . We also study lifetime asymptotics of Brownian-time Brownian motion, Z¹_t = z+X(|Y(t)|), 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.

Mathematics Subject Classification. 60J65, 60K99.

Key words: Iterated Brownian motion, Brownian-time Brownian motion, exit time, bounded domain, twisted domain, unbounded convex domain.

© EDP Sciences, SMAI 2007