Lifetime asymptotics of iterated Brownian motion in ℝ n

Erkan Nane

ESAIM: Probability and Statistics (2007)

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

Abstract

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Let τ D ( Z ) be the first exit time of iterated Brownian motion from a domain D ⊂ ℝ n started at z ∈ 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 DeBlassie (2004) [DeBlassie, Ann. Appl. Prob.14 (2004) 1529–1558] and Nane (2006) [Nane, Stochastic Processes Appl.116 (2006) 905–916], for z ∈ D 
 lim t → ∞ t - 1 / 2 exp 3 2 π 2 / 3 λ D 2 / 3 t 1 / 3 P z [ τ D ( Z ) > t ] = C ( z ) , 
where C ( z ) = ( λ D 2 7 / 2 ) / 3 π ψ ( z ) ∫ D ψ ( y ) d y 2 . Here λD is the first eigenvalue of the Dirichlet Laplacian 1 2 Δ in D, and ψ is the eigenfunction corresponding to λD. We also study lifetime asymptotics of Brownian-time Brownian motion, Z t 1 = 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.

How to cite

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Nane, 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

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  1. H. Allouba, Brownian-time processes: The pde connection and the corresponding Feynman-Kac formula. Trans. Amer. Math. Soc.354 (2002) 4627–4637.  
  2. H. Allouba and W. Zheng, Brownian-time processes: The pde connection and the half-derivative generator. Ann. Prob.29 (2001) 1780–1795.  
  3. R. Bañuelos and R.D. DeBlassie, The exit distribution for iterated Brownian motion in cones. Stochastic Processes Appl.116 (2006) 36–69.  
  4. 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.  
  5. R. Bañuelos, R. Smits, Brownian motion in cones. Probab. Theory Relat. Fields108 (1997) 299–319.  
  6. N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation. Cambridge University Press, Cambridge (1987).  
  7. 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.  
  8. 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.  
  9. K. Burdzy and D. Khoshnevisan, Brownian motion in a Brownian crack. Ann. Appl. Probabl.8 (1998) 708–748.  
  10. E. Csàki, M. Csörgő, A. Földes and P. Révész, The local time of iterated Brownian motion. J. Theoret. Probab.9 (1996) 717–743.  
  11. R.D. DeBlassie, Exit times from cones in ℝ n of Brownian motion. Prob. Th. Rel. Fields74 (1987) 1–29.  
  12. R.D. DeBlassie, Iterated Brownian motion in an open set. Ann. Appl. Prob.14 (2004) 1529–1558.  
  13. R.D. DeBlassie and R. Smits, Brownian motion in twisted domains. Trans. Amer. Math. Soc.357 (2005) 1245–1274.  
  14. N.G. De Bruijn, Asymptotic methods in analysis. North-Holland Publishing Co., Amsterdam (1957).  
  15. N. Eisenbum and Z. Shi, Uniform oscillations of the local time of iterated Brownian motion. Bernoulli5 (1999) 49–65.  
  16. W. Feller, An Introduction to Probability Theory and its Applications. Wiley, New York (1971).  
  17. Y. Kasahara, Tauberian theorems of exponential type. J. Math. Kyoto Univ.12 (1978) 209–219.  
  18. D. Khoshnevisan and T.M. Lewis, Stochastic calculus for Brownian motion in a Brownian fracture. Ann. Applied Probabl. 9 (1999) 629–667.  
  19. 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.  
  20. O. Laporte, Absorption coefficients for thermal neutrons. Phys. Rev.52 (1937) 72–74.  
  21. W. Li, The first exit time of a Brownian motion from an unbounded convex domain. Ann. Probab.31 (2003) 1078–1096.  
  22. M. Lifshits and Z. Shi, The first exit time of Brownian motion from a parabolic domain. Bernoulli8 (2002) 745–765.  
  23. E. Nane, Iterated Brownian motion in parabola-shaped domains. Potential Analysis24 (2006) 105–123.  
  24. E. Nane, Iterated Brownian motion in bounded domains in ℝ n . Stochastic Processes Appl.116 (2006) 905–916.  
  25. E. Nane, Higher order PDE's and iterated processes. Accepted Trans. Amer. Math. Soc. math.PR/0508262.  
  26. E. Nane, Laws of the iterated logarithm for α-time Brownian motion. Electron. J. Probab.11 (2006) 34–459 (electronic).  
  27. E. Nane, Isoperimetric-type inequalities for iterated Brownian motion in ℝ n . Submitted, math.PR/0602188.  
  28. S.C. Port and C.J. Stone, Brownian motion and Classical potential theory. Academic, New York (1978).  
  29. Y. Xiao, Local times and related properties of multidimensional iterated Brownian motion. J. Theoret. Probab.11 (1998) 383–408.  

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