Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Asymptotics of the densities of the first passage time distributions for Bessel diffusions

Author: Kôhei Uchiyama
Journal: Trans. Amer. Math. Soc. 367 (2015), 2719-2742
MSC (2010): Primary 60J65; Secondary 60J60
Published electronically: September 4, 2014
MathSciNet review: 3301879
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper concerns the first passage times to a point $ a >0$, denoted by $ \sigma _a$, of Bessel processes. We are interested in the case when the process starts at $ x>a$ and we compute the densities of the distributions of $ \sigma _a$ to obtain the exact asymptotic forms of them as $ t\to \infty $ that are valid uniformly in $ x>a$ for every order of the Bessel process.

References [Enhancements On Off] (What's this?)

  • [1] T. Byczkowski and M. Ryznar, Hitting distributions of geometric Brownian motion, Studia Math. 173 (2006), no. 1, 19-38. MR 2204460 (2007e:60082),
  • [2] T. Byczkowski, J. Małecki, and M. Ryznar, Hitting Times of Bessel Processes, Potential Anal. 38 (2013), no. 3, 753-786. MR 3034599,
  • [3] Z. Ciesielski and S. J. Taylor, First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path, Trans. Amer. Math. Soc. 103 (1962), 434-450. MR 0143257 (26 #816)
  • [4] A. Erdélyi, Asymptotic expansions, Dover Publications Inc., New York, 1956. MR 0078494 (17,1202c)
  • [5] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. Based, in part, on notes left by Harry Bateman. MR 0061695 (15,868a)
  • [6] Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845 (99e:35001)
  • [7] Alexander Grigor'yan and Laurent Saloff-Coste, Hitting probabilities for Brownian motion on Riemannian manifolds, J. Math. Pures Appl. (9) 81 (2002), no. 2, 115-142. MR 1994606 (2005e:31011),
  • [8] R. K. Getoor and M. J. Sharpe, Excursions of Brownian motion and Bessel processes, Z. Wahrsch. Verw. Gebiete 47 (1979), no. 1, 83-106. MR 521534 (80b:60104),
  • [9] Y. Hamana and H. Matumoto, The probability distributions of the first hitting times of Bessel processes, Trans. Amer. Math. Soc. 365 (2013), no. 10, 5237-5257. MR 3074372
  • [10] Y. Hamana and H. Matumoto, The probability densities of the first hitting times of Bessel processes, J. Math-for-Ind. 4B (2012), 91-95. MR 3072321
  • [11] K. Itô and H.P. McKean, Jr., Diffusion processes and their sample paths, Springer, 1965. MR 0199891
  • [12] John T. Kent, Eigenvalue expansions for diffusion hitting times, Z. Wahrsch. Verw. Gebiete 52 (1980), no. 3, 309-319. MR 576891 (81i:60072),
  • [13] N. N. Lebedev, Special functions and their applications, Revised English edition. Translated and edited by Richard A. Silverman, Prentice-Hall Inc., Englewood Cliffs, N.J., 1965. MR 0174795 (30 #4988)
  • [14] Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357 (2000h:60050)
  • [15] Frank Spitzer, Some theorems concerning $ 2$-dimensional Brownian motion, Trans. Amer. Math. Soc. 87 (1958), 187-197. MR 0104296 (21 #3051)
  • [16] E. C. Titchmarsh, Eigenfunction expansions associated with second-order differential equations. Part I, Second Edition, Clarendon Press, Oxford, 1962. MR 0176151 (31 #426)
  • [17] Kôhei Uchiyama, The first hitting time of a single point for random walks, Electron. J. Probab. 16 (2011), no. 71, 1960-2000. MR 2851052 (2012m:60107),
  • [18] Kôhei Uchiyama, Asymptotic estimates of the distribution of Brownian hitting time of a disc, J. Theoret. Probab. 25 (2012), no. 2, 450-463. MR 2914437,
  • [19] Kôhei Uchiyama, The expected area of the Wiener sausage swept by a disc, Stochastic Process. Appl. 123 (2013), no. 1, 191-211. MR 2988115,
  • [20] K. Uchiyama, The expected volume of Wiener sausage for Brownian bridge joining the origin to a point outside a parabolic region, RIMS Kôkyûroku (2013).
  • [21] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR 1349110 (96i:33010)
  • [22] Neil A. Watson, Introduction to heat potential theory, Mathematical Surveys and Monographs, vol. 182, American Mathematical Society, Providence, RI, 2012. MR 2907452

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 60J65, 60J60

Retrieve articles in all journals with MSC (2010): 60J65, 60J60

Additional Information

Kôhei Uchiyama
Affiliation: Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro Tokyo 152-8551, Japan

Keywords: First passage time, exterior problem, uniform estimate, Bessel diffusion
Received by editor(s): August 3, 2012
Received by editor(s) in revised form: March 11, 2013
Published electronically: September 4, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society