Lifetime asymptotics of iterated Brownian motion in R^{n}

Abstract

Let $\tau_{D}(Z) $ be the first exit time of iterated Brownian motion from a domain $D \subset \RR{R}^{n}$ started at $z\in D$ and let $P_{z}[\tau_{D}(Z) >t]$ be its distribution. In this paper we establish the exact asymptotics of $P_{z}[\tau_{D}(Z) >t]$ over bounded domains as an improvement of the results in \cite{deblassie, nane2}, for $z\in D$ \begin{eqnarray} \lim_{t\to\infty} t^{-1/2}\exp({3/2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3}) P_{z}[\tau_{D}(Z)>t]= C(z),\nonumber \end{eqnarray} where $C(z)=(\lambda_{D}2^{7/2})/\sqrt{3 \pi}(\psi(z)\int_{D}\psi(y)dy) ^{2}$. Here $\lambda_{D}$ is the first eigenvalue of the Dirichlet Laplacian ${1/2}\Delta$ in $D$, and $\psi $ is the eigenfunction corresponding to $\lambda_{D}$ . We also study lifetime asymptotics of Brownian-time Brownian motion (BTBM), $Z^{1}_{t}=z+X(|Y(t)|)$, where $X_{t}$ and $Y_{t}$ are independent one-dimensional Brownian motions.