Iterated Brownian motion in bounded domains in R^n

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 $z\in D$ $$P_{z}[\tau_{D}(Z)>t]\approx t^{1/2} \exp(-{3/2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3}), as t\to\infty .$$ We also study asymptotics of the life time of Brownian-time Brownian motion (BTBM), $Z^{1}_{t}=z+X(Y(t))$, where $X_{t}$ and $Y_{t}$ are independent one-dimensional Brownian motions.