# Lifetime asymptotics of iterated Brownian motion in ${\mathbb{R}}^{n}$

ESAIM: Probability and Statistics (2007)

- Volume: 11, page 147-160
- ISSN: 1292-8100

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topNane, Erkan. "Lifetime asymptotics of iterated Brownian motion in $\mathbb{R}^{n}$." ESAIM: Probability and Statistics 11 (2007): 147-160. <http://eudml.org/doc/250092>.

@article{Nane2007,

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.
},

author = {Nane, Erkan},

journal = {ESAIM: Probability and Statistics},

keywords = {Iterated Brownian motion;
Brownian-time Brownian motion; exit time; bounded domain; twisted
domain; unbounded convex domain.; iterated Brownian motion; Brownian-time Brownian motion; twisted domain; unbounded convex domain},

language = {eng},

month = {3},

pages = {147-160},

publisher = {EDP Sciences},

title = {Lifetime asymptotics of iterated Brownian motion in $\mathbb\{R\}^\{n\}$},

url = {http://eudml.org/doc/250092},

volume = {11},

year = {2007},

}

TY - JOUR

AU - Nane, Erkan

TI - Lifetime asymptotics of iterated Brownian motion in $\mathbb{R}^{n}$

JO - ESAIM: Probability and Statistics

DA - 2007/3//

PB - EDP Sciences

VL - 11

SP - 147

EP - 160

AB -
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.

LA - eng

KW - Iterated Brownian motion;
Brownian-time Brownian motion; exit time; bounded domain; twisted
domain; unbounded convex domain.; iterated Brownian motion; Brownian-time Brownian motion; twisted domain; unbounded convex domain

UR - http://eudml.org/doc/250092

ER -

## References

top- H. Allouba, Brownian-time processes: The pde connection and the corresponding Feynman-Kac formula. Trans. Amer. Math. Soc.354 (2002) 4627–4637. Zbl1006.60063
- H. Allouba and W. Zheng, Brownian-time processes: The pde connection and the half-derivative generator. Ann. Prob.29 (2001) 1780–1795. Zbl1018.60066
- R. Bañuelos and R.D. DeBlassie, The exit distribution for iterated Brownian motion in cones. Stochastic Processes Appl.116 (2006) 36–69. Zbl1085.60058
- R. Bañuelos, R.D. DeBlassie and R. Smits, The first exit time of planar Brownian motion from the interior of a parabola. Ann. Prob.29 (2001) 882–901. Zbl1013.60060
- R. Bañuelos, R. Smits, Brownian motion in cones. Probab. Theory Relat. Fields108 (1997) 299–319. Zbl0884.60037
- N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation. Cambridge University Press, Cambridge (1987). Zbl0617.26001
- K. Burdzy, Some path properties of iterated Brownian motion, in Seminar on Stochastic Processes, E. Çinlar, K.L. Chung and M.J. Sharpe, Eds., Birkhäuser, Boston (1993) 67–87. Zbl0789.60060
- K. Burdzy, Variation of iterated Brownian motion, in Workshops and Conference on Measure-valued Processes, Stochastic Partial Differential Equations and Interacting Particle Systems, D.A. Dawson, Ed., Amer. Math. Soc. Providence, RI (1994) 35–53. Zbl0803.60077
- K. Burdzy and D. Khoshnevisan, Brownian motion in a Brownian crack. Ann. Appl. Probabl.8 (1998) 708–748. Zbl0937.60081
- E. Csàki, M. Csörgő, A. Földes and P. Révész, The local time of iterated Brownian motion. J. Theoret. Probab.9 (1996) 717–743. Zbl0857.60081
- R.D. DeBlassie, Exit times from cones in ${\mathbb{R}}^{n}$ of Brownian motion. Prob. Th. Rel. Fields74 (1987) 1–29. Zbl0586.60077
- R.D. DeBlassie, Iterated Brownian motion in an open set. Ann. Appl. Prob.14 (2004) 1529–1558. Zbl1051.60082
- R.D. DeBlassie and R. Smits, Brownian motion in twisted domains. Trans. Amer. Math. Soc.357 (2005) 1245–1274. Zbl1061.60084
- N.G. De Bruijn, Asymptotic methods in analysis. North-Holland Publishing Co., Amsterdam (1957). Zbl0098.26404
- N. Eisenbum and Z. Shi, Uniform oscillations of the local time of iterated Brownian motion. Bernoulli5 (1999) 49–65. Zbl0930.60056
- W. Feller, An Introduction to Probability Theory and its Applications. Wiley, New York (1971). Zbl0219.60003
- Y. Kasahara, Tauberian theorems of exponential type. J. Math. Kyoto Univ.12 (1978) 209–219. Zbl0421.40009
- D. Khoshnevisan and T.M. Lewis, Stochastic calculus for Brownian motion in a Brownian fracture. Ann. Applied Probabl. 9 (1999) 629–667. Zbl0956.60054
- D. Khoshnevisan and T.M. Lewis, Chung's law of the iterated logarithm for iterated Brownian motion. Ann. Inst. H. Poincaré Probab. Statist.32 (1996) 349–359. Zbl0859.60025
- O. Laporte, Absorption coefficients for thermal neutrons. Phys. Rev.52 (1937) 72–74. Zbl63.1399.01
- W. Li, The first exit time of a Brownian motion from an unbounded convex domain. Ann. Probab.31 (2003) 1078–1096. Zbl1030.60032
- M. Lifshits and Z. Shi, The first exit time of Brownian motion from a parabolic domain. Bernoulli8 (2002) 745–765. Zbl1018.60084
- E. Nane, Iterated Brownian motion in parabola-shaped domains. Potential Analysis24 (2006) 105–123. Zbl1090.60071
- E. Nane, Iterated Brownian motion in bounded domains in ${\mathbb{R}}^{n}$. Stochastic Processes Appl.116 (2006) 905–916. Zbl1106.60309
- E. Nane, Higher order PDE's and iterated processes. Accepted Trans. Amer. Math. Soc. math.PR/0508262. Zbl1157.60071
- E. Nane, Laws of the iterated logarithm for α-time Brownian motion. Electron. J. Probab.11 (2006) 34–459 (electronic). Zbl1121.60085
- E. Nane, Isoperimetric-type inequalities for iterated Brownian motion in ${\mathbb{R}}^{n}$. Submitted, math.PR/0602188. Zbl1134.60051
- S.C. Port and C.J. Stone, Brownian motion and Classical potential theory. Academic, New York (1978). Zbl0413.60067
- Y. Xiao, Local times and related properties of multidimensional iterated Brownian motion. J. Theoret. Probab.11 (1998) 383–408. Zbl0914.60063

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