Abstract
Let \(T_1^{(\mu)}\) be the first hitting time of the point 1 by the Bessel process with index μ ∈ ℝ starting from x > 1. Using an integral formula for the density \(q_x^{(\mu)}(t)\) of \(T_1^{(\mu)}\), obtained in Byczkowski and Ryznar (Stud Math 173(1):19–38, 2006), we prove sharp estimates of the density of \(T_1^{(\mu)}\) which exhibit the dependence both on time and space variables. Our result provides optimal uniform estimates for the density of the hitting time of the unit ball by the Brownian motion in ℝn, which improve existing bounds. Another application is to provide sharp estimates for the Poisson kernel for half-spaces for hyperbolic Brownian motion in real hyperbolic spaces.
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References
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th edn. Dover, New York (1972)
Borodin, A.N., Salminen, P.: Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhauser, Basel (2002)
Byczkowski, T., Graczyk, P., Stos, A.: Poisson kernels of half-spaces in real hyperbolic spaces. Rev. Mat. Iberoam. 23(1), 85–126 (2007)
Byczkowski, T., Małecki, J., Ryznar, M.: Hitting Half-spaces by Bessel–Brownian diffusions. Potential Anal. 33, 47–83 (2010). arXiv:math.PR/0612176
Byczkowski, T., Ryznar, M.: Hitting distibution of geometric Brownian motion. Stud. Math. 173(1), 19–38 (2006)
Collete, P., Martinez, S., San Martin, J.: Asymptotic behaviour of a Brownian motion on exterior domains. Probab. Theory Relat. Fields 116, 303–316 (2000)
Davies, E.B.: Heat kernels and spectral theory. In: Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1990)
Erdelyi, A., et al.: Tables of Integral Transforms, vols. I and II. McGraw-Hill, New York (1954)
Getoor, R.K., Sharpe, M.J.: Excursions of Brownian motion and Bessel processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 47, 83–106 (1979)
Grigor’yan, A., Saloff-Coste, L.: Hitting probabilities for Brownian motion on Riemannian manifolds. J. Math. Pure Appl. 81, 115–142 (2002)
Gruet, J.-C.: Semi-groupe du mouvement Brownien hyperbolique. Stoch. Stoch. Rep. 56, 53–61 (1996)
Hunt, G.A.: Some theorems concerning brownian motion. Trans. Am. Math. Soc. 81, 294–319 (1956)
Itô, K., McKean Jr., H.P.: Diffusion Processes and Their Sample Paths. Springer, New York (1974)
Kent, J.T.: Some probabilistic properties of Bessel functions. Ann. Probab. 6, 760–770 (1978)
Lamperti, J.: Semi-stable Markov processes. Z. Wahrscheinlichkeitstheor. Verw. Geb. 22, 205–225 (1972)
Matsumoto, H., Yor, M.: Exponential functionals of Brownian motion, I: probability laws at fixed time. Probab. Surv. 2, 312–347 (2005)
Matsumoto, H., Yor, M.: Exponential functionals of Brownian motion, II: some related diffusion processes. Probab. Surv. 2, 348–384 (2005)
McKean, H.P.: The Bessel motion and a singular integral equation. Mem. Sci. Univ. Kyoto, Ser A, Math. 33, 317–322 (1960)
Molchanov, S.A., Ostrowski, S.A.: Symmetric stable processes as traces of degenerate diffusion processes. Theory Probab. Appl. 12, 128–131 (1969)
Port, S.C.: Hitting times for transient stable process. Pac. J. Math. 21, 161–165 (1967)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, New York (1999)
Uchiyama, K.: Asymptotic estimates of the distribution of brownian hitting time of a disc. J. Theor. Probab. 25(2), 450–463 (2012)
Yor, M.: Some Aspects of Brownian Motion, Part I: Some Special Functional. Birkhaäuser, Basel (1992)
Yor, M.: Exponential Functionals of Brownian Motion and Related Processes. Springer Finance, Berlin (2001)
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Research supported by Polish Ministry of Science and Higher Eduction grant N N201 3731 36.
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Byczkowski, T., Małecki, J. & Ryznar, M. Hitting Times of Bessel Processes. Potential Anal 38, 753–786 (2013). https://doi.org/10.1007/s11118-012-9296-7
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DOI: https://doi.org/10.1007/s11118-012-9296-7