Lifetime asymptotics of iterated Brownian motion in n

Erkan Nane

ESAIM: Probability and Statistics (2007)

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

Abstract

top
Let τ D ( Z ) be the first exit time of iterated Brownian motion from a domain D 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

top

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

top
  1. H. Allouba, Brownian-time processes: The pde connection and the corresponding Feynman-Kac formula. Trans. Amer. Math. Soc.354 (2002) 4627–4637.  Zbl1006.60063
  2. H. Allouba and W. Zheng, Brownian-time processes: The pde connection and the half-derivative generator. Ann. Prob.29 (2001) 1780–1795.  Zbl1018.60066
  3. R. Bañuelos and R.D. DeBlassie, The exit distribution for iterated Brownian motion in cones. Stochastic Processes Appl.116 (2006) 36–69.  Zbl1085.60058
  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.  Zbl1013.60060
  5. R. Bañuelos, R. Smits, Brownian motion in cones. Probab. Theory Relat. Fields108 (1997) 299–319.  Zbl0884.60037
  6. N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation. Cambridge University Press, Cambridge (1987).  Zbl0617.26001
  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.  Zbl0789.60060
  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.  Zbl0803.60077
  9. K. Burdzy and D. Khoshnevisan, Brownian motion in a Brownian crack. Ann. Appl. Probabl.8 (1998) 708–748.  Zbl0937.60081
  10. 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
  11. R.D. DeBlassie, Exit times from cones in n of Brownian motion. Prob. Th. Rel. Fields74 (1987) 1–29.  Zbl0586.60077
  12. R.D. DeBlassie, Iterated Brownian motion in an open set. Ann. Appl. Prob.14 (2004) 1529–1558.  Zbl1051.60082
  13. R.D. DeBlassie and R. Smits, Brownian motion in twisted domains. Trans. Amer. Math. Soc.357 (2005) 1245–1274.  Zbl1061.60084
  14. N.G. De Bruijn, Asymptotic methods in analysis. North-Holland Publishing Co., Amsterdam (1957).  Zbl0098.26404
  15. N. Eisenbum and Z. Shi, Uniform oscillations of the local time of iterated Brownian motion. Bernoulli5 (1999) 49–65.  Zbl0930.60056
  16. W. Feller, An Introduction to Probability Theory and its Applications. Wiley, New York (1971).  Zbl0219.60003
  17. Y. Kasahara, Tauberian theorems of exponential type. J. Math. Kyoto Univ.12 (1978) 209–219.  Zbl0421.40009
  18. D. Khoshnevisan and T.M. Lewis, Stochastic calculus for Brownian motion in a Brownian fracture. Ann. Applied Probabl. 9 (1999) 629–667.  Zbl0956.60054
  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.  Zbl0859.60025
  20. O. Laporte, Absorption coefficients for thermal neutrons. Phys. Rev.52 (1937) 72–74.  Zbl63.1399.01
  21. W. Li, The first exit time of a Brownian motion from an unbounded convex domain. Ann. Probab.31 (2003) 1078–1096.  Zbl1030.60032
  22. M. Lifshits and Z. Shi, The first exit time of Brownian motion from a parabolic domain. Bernoulli8 (2002) 745–765.  Zbl1018.60084
  23. E. Nane, Iterated Brownian motion in parabola-shaped domains. Potential Analysis24 (2006) 105–123.  Zbl1090.60071
  24. E. Nane, Iterated Brownian motion in bounded domains in n . Stochastic Processes Appl.116 (2006) 905–916.  Zbl1106.60309
  25. E. Nane, Higher order PDE's and iterated processes. Accepted Trans. Amer. Math. Soc. math.PR/0508262.  Zbl1157.60071
  26. E. Nane, Laws of the iterated logarithm for α-time Brownian motion. Electron. J. Probab.11 (2006) 34–459 (electronic).  Zbl1121.60085
  27. E. Nane, Isoperimetric-type inequalities for iterated Brownian motion in n . Submitted, math.PR/0602188.  Zbl1134.60051
  28. S.C. Port and C.J. Stone, Brownian motion and Classical potential theory. Academic, New York (1978).  Zbl0413.60067
  29. Y. Xiao, Local times and related properties of multidimensional iterated Brownian motion. J. Theoret. Probab.11 (1998) 383–408.  Zbl0914.60063

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.