## Electronic Journal of Probability

### On the exit time from a cone for brownian motion with drift

#### Abstract

We investigate the tail distribution of the first exit time of Brownian motion with drift from a cone and find its exact asymptotics for a large class of cones. Our results show in particular that its exponential decreasing rate is a function of the distance between the drift and the cone, whereas the polynomial part in the asymptotics depends on the position of the drift with respect to the cone and its polar cone, and reflects the local geometry of the cone at the points that minimize the distance to the drift.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 63, 27 pp.

Dates
Accepted: 20 July 2014
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465065705

Digital Object Identifier
doi:10.1214/EJP.v19-3169

Mathematical Reviews number (MathSciNet)
MR3238783

Zentralblatt MATH identifier
1317.60104

Rights

#### Citation

Garbit, Rodolphe; Raschel, Kilian. On the exit time from a cone for brownian motion with drift. Electron. J. Probab. 19 (2014), paper no. 63, 27 pp. doi:10.1214/EJP.v19-3169. https://projecteuclid.org/euclid.ejp/1465065705

#### References

• Bañuelos, Rodrigo; Smits, Robert G. Brownian motion in cones. Probab. Theory Related Fields 108 (1997), no. 3, 299–319.
• Biane, Philippe. Quelques propriétés du mouvement brownien dans un cone. (French) [Some properties of Brownian motion in a cone] Stochastic Process. Appl. 53 (1994), no. 2, 233–240.
• Biane, Philippe. Permutation model for semi-circular systems and quantum random walks. Pacific J. Math. 171 (1995), no. 2, 373–387.
• Biane, Philippe; Bougerol, Philippe; O'Connell, Neil. Littelmann paths and Brownian paths. Duke Math. J. 130 (2005), no. 1, 127–167.
• Burkholder, D. L. Exit times of Brownian motion, harmonic majorization, and Hardy spaces. Advances in Math. 26 (1977), no. 2, 182–205.
• Chavel, Isaac. Eigenvalues in Riemannian geometry. Including a chapter by Burton Randol. With an appendix by Jozef Dodziuk. Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984. xiv+362 pp. ISBN: 0-12-170640-0
• Copson, E. T. Asymptotic expansions. Cambridge Tracts in Mathematics and Mathematical Physics, No. 55 Cambridge University Press, New York 1965 vii+120 pp.
• DeBlassie, R. Dante. Exit times from cones in ${\bf R}^ n$ of Brownian motion. Probab. Theory Related Fields 74 (1987), no. 1, 1–29.
• DeBlassie, R. Dante. Remark on: "Exit times from cones in ${\bf R}^ n$ of Brownian motion” [Probab. Theory Related Fields 74 (1987), no. 1, 1?29; (88d:60205)]. Probab. Theory Related Fields 79 (1988), no. 1, 95–97.
• Doumerc, Yan; O'Connell, Neil. Exit problems associated with finite reflection groups. Probab. Theory Related Fields 132 (2005), no. 4, 501–538.
• Dyson, Freeman J. A Brownian-motion model for the eigenvalues of a random matrix. J. Mathematical Phys. 3 1962 1191–1198.
• Friedman, Avner. Remarks on the maximum principle for parabolic equations and its applications. Pacific J. Math. 8 1958 201–211.
• Garbit, Rodolphe. Brownian motion conditioned to stay in a cone. J. Math. Kyoto Univ. 49 (2009), no. 3, 573–592.
• Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 224. Springer-Verlag, Berlin, 1983. xiii+513 pp. ISBN: 3-540-13025-X
• Grabiner, David J. Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. H. PoincarÃ© Probab. Statist. 35 (1999), no. 2, 177–204.
• Hunt, G. A. Some theorems concerning Brownian motion. Trans. Amer. Math. Soc. 81 (1956), 294–319.
• Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8
• Karlin, Samuel; McGregor, James. Coincidence probabilities. Pacific J. Math. 9 1959 1141–1164.
• Kinderlehrer, David; Nirenberg, Louis. Analyticity at the boundary of solutions of nonlinear second-order parabolic equations. Comm. Pure Appl. Math. 31 (1978), no. 3, 283–338.
• KÃ¶nig, W.; Schmid, P. Brownian motion in a truncated Weyl chamber. Markov Process. Related Fields 17 (2011), no. 4, 499–522.
• Meyre, Thierry. Étude asymptotique du temps passé par le mouvement brownien dans un cône. (French) [Asymptotic study of the sojourn time of Brownian motion in a cone] Ann. Inst. H. PoincarÃ© Probab. Statist. 27 (1991), no. 1, 107–124.
• Morrey, C. B., Jr.; Nirenberg, L. On the analyticity of the solutions of linear elliptic systems of partial differential equations. Comm. Pure Appl. Math. 10 (1957), 271–290.
• Puchała, Zbigniew; Rolski, Tomasz. The exact asymptotic of the collision time tail distribution for independent Brownian particles with different drifts. Probab. Theory Related Fields 142 (2008), no. 3-4, 595–617.
• Spitzer, Frank. Some theorems concerning $2$-dimensional Brownian motion. Trans. Amer. Math. Soc. 87 1958 187–197.
• Watson, G. N. A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. vi+804 pp.