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

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• Stochastic solutions of a class of Higher order Cauchy problems in $\rd$
• Lifetime asymptotics of iterated Brownian motion in R^{n}
• Isoperimetric-type inequalities for iterated Brownian motion in R^n