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

Abstract

Let $τ_{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 $P_{z}[τ_{D}(Z) >t]$ be its distribution. In this paper we establish the exact asymptotics of $P_{z}[τ_{D}(Z) >t]$ over bounded domains as an improvement of the results in \cite{deblassie, nane2}, for $z\in D$ \begin{eqnarray} \lim_{t\to\infty} t^{-1/2}\exp({3/2}π^{2/3}λ_{D}^{2/3}t^{1/3}) P_{z}[τ_{D}(Z)>t]= C(z),\nonumber \end{eqnarray} where $C(z)=(λ_{D}2^{7/2})/\sqrt{3 π}(ψ(z)\int_{D}ψ(y)dy) ^{2}$. Here $λ_{D}$ is the first eigenvalue of the Dirichlet Laplacian ${1/2}Δ$ in $D$, and $ψ$ is the eigenfunction corresponding to $λ_{D}$ . 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.