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 $D\subset \text{RR}{R}^n$ started at $z\in D$ and let $P_{z}[\tau_{D}(Z)

t]$beitsdistribution\mathrm{.}Inthispaperweestablishtheexactasymptoticsof$P_{z}[\tau_{D}(Z) >t]$ over bounded domains as an extension of the result in DeBlassie \cite{deblassie}, for $z\in D$ $$P_z[{\tau }_D(Z)>t]\approx t^{1\mathrm{/}2}\exp (-3\mathrm{/}2{\pi }^{2\mathrm{/}3}{\lambda }_D^{2\mathrm{/}3}{t}^{1\mathrm{/}3}),ast\to \mathrm{\infty }\mathrm{.}$$ 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.