Iterated Brownian motion in bounded domains in R^n

Abstract

Let ${\tau }_D(Z)$ is the first exit time of iterated Brownian motion from a domain $D \subset \RR{R}^{n}$ 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 extension of the result in DeBlassie \cite{deblassie}, for $z\in D$ $$P_z[{\tau }_D(Z)>t]\approx t^{1/2}\exp (-3/2{\pi }^{2/3}{\lambda }_D^{2/3}{t}^{1/3}),ast\to \infty .$$ We also study asymptotics of the life time of Brownian-time Brownian motion (BTBM), $Z_t^1=z+X(Y(t))$, where $X_t$ and $Y_t$ are independent one-dimensional Brownian motions.

17 pages

References in corpus (3)

• Iterated Brownian motion in an open set
• Iterated Brownian Motion in Parabola-Shaped Domains
• The exit distribution for iterated Brownian motion in cones

Cited by in corpus (1)

• Laws of the iterated logarithm for α-time Brownian motion