| In Chapter 1, iterated Brownian motion started at z∈R is defined by Zt=z+XYt ,t≥0, where Xt= X+t,t≥0 X--t ,t<0 is a two-sided Brownian motion and X+t,X-t and Yt are three independent one-dimensional Brownian motions, all started at 0. In Rn one requires X+/- to be independent n-dimensional Brownian motions. Let tauD( Z) be the first exit time of this processes from a domain D ⊂ Rn , started at z ∈ D. In Chapter 2, we establish sub-exponential decay of large time asymptotics of P z[tauD(Z) > t] for several unbounded domains including parabola-shaped domains of the form Palpha = {(x, Y) ∈ RxRn-1:x>0, Y |
