Iterated Brownian Motion in Parabola-Shaped Domains

Iterated Brownian motion Z_t serves as a physical model for diffusions in a crack. If τ_D(Z) is the first exit time of this processes from a domain D⊂ℝⁿ, started at z∈D, then P_z[τ_D(Z)>t] is the distribution of the lifetime of the process in D. In this paper we determine the large time asymptotics of which gives exponential integrability of for parabola-shaped domains of the form P_α={(x,Y)∈ℝ×ℝⁿ⁻¹:x>0, |Y|<Ax^α}, for 0<α<1, A>0. We also obtain similar results for twisted domains in ℝ² as defined in DeBlassie and Smits: Brownian motion in twisted domains, Preprint, 2004. In particular, for a planar iterated Brownian motion in a parabola we find that for z∈℘ Mathematics Subject Classifications (2000)  60J65, 60K99. Erkan Nane: Supported in part by NSF Grant # 9700585-DMS.