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Lifetime asymptotics of iterated Brownian motion in $\mathbb{R}^{n}$![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20161010045614234-0760:S1292810007000122:S1292810007000122_eqnU1.gif)
Published online by Cambridge University Press: 31 March 2007
Abstract
Let $\tau _{D}(Z) $ be the first exit time of
iterated Brownian motion from a domain $D \subset \mathbb{R}^{n}$
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
DeBlassie (2004) [DeBlassie, Ann. Appl. Prob.14 (2004) 1529–1558] and Nane (2006) [Nane, Stochastic Processes Appl.116
(2006) 905–916], for $z\in D$
$ \displaystyle \lim_{t\to\infty}
t^{-1/2}\exp\left(\frac{3}{2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3}\right)
P_{z}[\tau_{D}(Z)>t]= C(z),\nonumber$
where $C(z)=(\lambda_{D}2^{7/2})/\sqrt{3 \pi}\left(
\psi(z)\int_{D}\psi(y){\rm d}y\right) ^{2}$
. Here λD is the
first eigenvalue of the Dirichlet Laplacian $\frac{1}{2}\Delta$
in
D, and ψ is the eigenfunction corresponding to
λD. We also study lifetime asymptotics of Brownian-time Brownian
motion,
$Z^{1}_{t} = z+X(|Y(t)|)$
, where Xt and Yt are independent
one-dimensional Brownian motions, in several unbounded domains. Using these results we obtain partial results for lifetime asymptotics of iterated Brownian motion in these unbounded domains.
Keywords
- Type
- Research Article
- Information
- Copyright
- © EDP Sciences, SMAI, 2007
References
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