Iterated Brownian Motion in Parabola-Shaped Domains

Iterated Brownian motion Zt serves as a physical model for diffusions in a crack. If τD(Z) is the first exit time of this processes from a domain D ⊂ Rn, started at z ∈ D, then Pz [τD(Z) > t] is the distribution of the lifetime of the process in D. In this paper we determine the large time asymptotics of Pz[τPα(Z) > t] which gives exponential integrability of τPα(Z) for parabola-shaped domains of the form Pα = {(x, Y ) ∈ R × Rn−1 : x > 0, |Y | < Axα}, for 0 < α < 1, A > 0. We also obtain similar results for twisted domains in R2 as defined in [9]. In particular, for a planar iterated Brownian motion in a parabola P = {(x, y) : x > 0, |y| < √ x} we find that for z ∈ P lim t→∞ t 1 7 logPz[τP(Z) > t] = − 7π2 225/7 . Mathematics Subject Classification (2000): 60J65, 60K99.