Iterated Brownian motion in bounded domains in RnRn

Let τD(Z)τD(Z) be the first exit time of iterated Brownian motion from a domain D⊂RnD⊂Rn started at z∈Dz∈D and let Pz[τD(Z)>t]Pz[τD(Z)>t] be its distribution. In this paper we establish the exact asymptotics of Pz[τD(Z)>t]Pz[τD(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∈Dz∈D: Pz[τD(Z)>t]≈t1/2exp(−32π2/3λD2/3t1/3),as t→∞. We also study asymptotics of the life time of Brownian-time Brownian motion (BTBM), Zt1=z+X(Y(t)), where XtXt and YtYt are independent one-dimensional Brownian motions.