Lifetime asymptotics of iterated brownian motion in ${ℝ}^n$

Nane, Erkan

ESAIM: Probability and Statistics, Tome 11 (2007), pp. 147-160

Résumé

Let ${\tau }_D\left(Z\right)$ be the first exit time of iterated brownian motion from a domain $D\subset {ℝ}^n$ started at $z\in D$ and let $P_z[{\tau }_D\left(Z\right)>t]$ be its distribution. In this paper we establish the exact asymptotics of $P_z[{\tau }_D\left(Z\right)>t]$ 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 $z\in D$ ${\displaystyle \underset{t\to \infty }{lim}{t}^{-1/2}exp\left(\frac{3}{2}{\pi }^{2/3}{\lambda }_D^{2/3}{t}^{1/3}\right)P_z[{\tau }_D\left(Z\right)>t]=C\left(z\right),}$ where $C\left(z\right)=\left({\lambda }_D{2}^{7/2}\right)/\sqrt{3\pi }{\left(\psi \left(z\right){\int }_D\psi \left(y\right)\mathrm{d}y\right)}^2$. Here ${\lambda }_D$ is the first eigenvalue of the Dirichlet laplacian $\frac{1}{2}\Delta$ in $D$, and $\psi$ is the eigenfunction corresponding to ${\lambda }_D$. We also study lifetime asymptotics of brownian-time brownian motion, $Z_t^1=z+X\left(\right|Y\left(t\right)\left|\right)$, 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.

Détail

DOI : 10.1051/ps:2007012

Classification : 60J65, 60K99
Keywords: iterated brownian motion, brownian-time brownian motion, exit time, bounded domain, twisted domain, unbounded convex domain