Lifetime asymptotics of iterated brownian motion in ${ℝ}^n$

Nane, Erkan

ESAIM: Probability and Statistics, Tome 11 (2007), pp. 147-160.

Let ${\tau }_D\left(Z\right)$ be the first exit time of iterated brownian motion from a domain $D\subset {ℝ}^n$ started at $z\in D$ and let $P_z[{\tau }_D\left(Z\right)>t]$ be its distribution. In this paper we establish the exact asymptotics of $P_z[{\tau }_D\left(Z\right)>t]$ 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 $z\in D$ ${\displaystyle \underset{t\to \infty }{lim}{t}^{-1/2}exp\left(\frac{3}{2}{\pi }^{2/3}{\lambda }_D^{2/3}{t}^{1/3}\right)P_z[{\tau }_D\left(Z\right)>t]=C\left(z\right),}$ where $C\left(z\right)=\left({\lambda }_D{2}^{7/2}\right)/\sqrt{3\pi }{\left(\psi \left(z\right){\int }_D\psi \left(y\right)\mathrm{d}y\right)}^2$. Here ${\lambda }_D$ is the first eigenvalue of the Dirichlet laplacian $\frac{1}{2}\Delta$ in $D$, and $\psi$ is the eigenfunction corresponding to ${\lambda }_D$. We also study lifetime asymptotics of brownian-time brownian motion, $Z_t^1=z+X\left(\right|Y\left(t\right)\left|\right)$, where $X_t$ and $Y_t$ 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.

[1] H. Allouba, Brownian-time processes: The pde connection and the corresponding Feynman-Kac formula. Trans. Amer. Math. Soc. 354 (2002) 4627-4637. | Zbl

[2] H. Allouba and W. Zheng, Brownian-time processes: The pde connection and the half-derivative generator. Ann. Prob. 29 (2001) 1780-1795. | Zbl

[3] R. Bañuelos and R.D. Deblassie, The exit distribution for iterated Brownian motion in cones. Stochastic Processes Appl. 116 (2006) 36-69. | Zbl

[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. | Zbl

[5] R. Bañuelos, R. Smits, Brownian motion in cones. Probab. Theory Relat. Fields 108 (1997) 299-319. | Zbl

[6] N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation. Cambridge University Press, Cambridge (1987). | MR | Zbl

[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. | Zbl

[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. | Zbl

[9] K. Burdzy and D. Khoshnevisan, Brownian motion in a Brownian crack. Ann. Appl. Probabl. 8 (1998) 708-748. | Zbl

[10] E. Csàki, M. Csörgő, A. Földes and P. Révész, The local time of iterated Brownian motion. J. Theoret. Probab. 9 (1996) 717-743. | Zbl

[11] R.D. Deblassie, Exit times from cones in ${ℝ}^n$ of Brownian motion. Prob. Th. Rel. Fields 74 (1987) 1-29. | Zbl

[12] R.D. Deblassie, Iterated Brownian motion in an open set. Ann. Appl. Prob. 14 (2004) 1529-1558. | Zbl

[13] R.D. Deblassie and R. Smits, Brownian motion in twisted domains. Trans. Amer. Math. Soc. 357 (2005) 1245-1274. | Zbl

[14] N.G. De Bruijn, Asymptotic methods in analysis. North-Holland Publishing Co., Amsterdam (1957). | Zbl

[15] N. Eisenbum and Z. Shi, Uniform oscillations of the local time of iterated Brownian motion. Bernoulli 5 (1999) 49-65. | Zbl

[16] W. Feller, An Introduction to Probability Theory and its Applications. Wiley, New York (1971). | MR | Zbl

[17] Y. Kasahara, Tauberian theorems of exponential type. J. Math. Kyoto Univ. 12 (1978) 209-219. | Zbl

[18] D. Khoshnevisan and T.M. Lewis, Stochastic calculus for Brownian motion in a Brownian fracture. Ann. Applied Probabl. 9 (1999) 629-667. | Zbl

[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. | Numdam | Zbl

[20] O. Laporte, Absorption coefficients for thermal neutrons. Phys. Rev. 52 (1937) 72-74. | JFM

[21] W. Li, The first exit time of a Brownian motion from an unbounded convex domain. Ann. Probab. 31 (2003) 1078-1096. | Zbl

[22] M. Lifshits and Z. Shi, The first exit time of Brownian motion from a parabolic domain. Bernoulli 8 (2002) 745-765. | Zbl

[23] E. Nane, Iterated Brownian motion in parabola-shaped domains. Potential Analysis 24 (2006) 105-123. | Zbl

[24] E. Nane, Iterated Brownian motion in bounded domains in ${ℝ}^n$. Stochastic Processes Appl. 116 (2006) 905-916. | Zbl

[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 $\alpha$-time Brownian motion. Electron. J. Probab. 11 (2006) 34-459 (electronic). | Zbl

[27] E. Nane, Isoperimetric-type inequalities for iterated Brownian motion in ${ℝ}^n$. Submitted, math.PR/0602188. | Zbl

[28] S.C. Port and C.J. Stone, Brownian motion and Classical potential theory. Academic, New York (1978). | MR | Zbl

[29] Y. Xiao, Local times and related properties of multidimensional iterated Brownian motion. J. Theoret. Probab. 11 (1998) 383-408. | Zbl