Iterated Brownian motion in bounded domains in

Let τ D ( Z ) be the first exit time of iterated Brownian motion from a domain D ⊂ R n started at z ∈ D and let P z [ τ D ( Z ) > t ] be its distribution. In this paper we establish the exact asymptotics of P z [ τ 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 ∈ D : P z [ τ D ( Z ) > t ] ≈ t 1 / 2 exp ( − 3 2 π 2 / 3 λ D 2 / 3 t 1 / 3 ) , as  t → ∞ . We also study asymptotics of the life time of Brownian-time Brownian motion (BTBM), Z t 1 = z + X ( Y ( t ) ) , where X t and Y t are independent one-dimensional Brownian motions.