Iterated Brownian motion in bounded domains in

Let [tau]D(Z) be the first exit time of iterated Brownian motion from a domain started at z[set membership, variant]D and let Pz[[tau]D(Z)>t] be its distribution. In this paper we establish the exact asymptotics of Pz[[tau]D(Z)>t] over bounded domains as an extension of the result in [R.D. DeBlassie, Iterated Brownian motion in an open set, Ann. Appl. Probab. 14 (3) (2004) 1529-1558], for z[set membership, variant]D: We also study asymptotics of the life time of Brownian-time Brownian motion (BTBM), , where Xt and Yt are independent one-dimensional Brownian motions.Iterated Brownian motion Brownian-time Brownian motion Exit time Bounded domain