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}π^{2/3}λ_{D}^{2/3}t^{1/3}) P_{z}[τ_{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), $Z_t^1=z+X(|Y(t)|)$, where $X_t$ and $Y_t$ are independent one-dimensional Brownian motions.

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• Brownian-Time Processes: The PDE Connection and the Half-Derivative Generator
• Iterated Brownian motion in an open set
• Brownian-Time Processes: The PDE Connection II and the Corresponding Feynman-Kac Formula
• Iterated Brownian Motion in Parabola-Shaped Domains
• Iterated Brownian motion in bounded domains in R^n
• Isoperimetric-type inequalities for iterated Brownian motion in R^n
• The exit distribution for iterated Brownian motion in cones

Cited by in corpus (5)

• Brownian subordinators and fractional Cauchy problems
• Large deviations for local time fractional Brownian motion and applications
• Laws of the iterated logarithm for a class of iterated processes
• Isoperimetric-type inequalities for iterated Brownian motion in R^n
• Spectral bounds for exit times on metric measure Dirichlet spaces and applications