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 $z\in \mathcal{ P}$ $$\lim_{ t\to\infty} t^{ -{ 1/7}} \log P_{ z}[\tau_{ \mathcal{ P}}(Z) >t]= - \frac{ 7 \pi ^{ 2}}{ 2^{ 25/ 7}}. $$
Categories: math.PR