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 $D \subset \RR{R}^{n}$, 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 $\RR{R}^{2}$ as defined in \cite{DSmits}. In particular, for a planar iterated Brownian motion in a parabola $\mathcal{P}=\{(x,y): x>0, |y|< \sqrt{x} \}$ we find that for $z\in \mathcal{P}$ $$\lim_{t\to\infty} t^{-{1/7}} \log P_{z}[\tau_{\mathcal{P}}(Z) >t]= - \frac{7 \pi ^{2}}{2^{25/ 7}}. $$