Iterated Brownian Motion in Parabola-Shaped Domains

Iterated Brownian motion $Z_t$ serves as a physical model for diffusions in a crack. If ${\tau }_D(Z)$ is the first exit time of this processes from a domain $D\subset \text{RR}{R}^n$, started at $z\in D$, then $P_z[{\tau }_D(Z)>t]$ is the distribution of the lifetime of the process in $D$. In this paper we determine the large time asymptotics of $P_z[{\tau }_{P_{\alpha }}(Z)>t]$ which gives exponential integrability of ${\tau }_{P_{\alpha }}(Z)$ for parabola-shaped domains of the form $ P_{\alpha}={(x,Y)\in \RR{R} \times \RR{R}^{n-1}: x>0, |Y|<Ax^{\alpha} }$,for$ 0<\alpha <1$,$A>0.$ We also obtain similar results for twisted domains in $\text{RR}{R}^2$ as defined in \cite{DSmits}. In particular, for a planar iterated Brownian motion in a parabola $$\mathcal{P}$={(x,y): x>0, |y|< \sqrt{x} }$wefindthatfor$z\in $\mathcal{P}$$ $$\underset{t\to \mathrm{\infty }}{\lim }{t}^{-1\mathrm{/}7}\log P_z[{\tau }_{\mathcal{P}}(Z)>t]=-\frac{7{\pi }^2}{2^{25\mathrm{/}7}}\mathrm{.}$$