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Chung's law for integrated Brownian motion


Authors: Davar Khoshnevisan and Zhan Shi
Journal: Trans. Amer. Math. Soc. 350 (1998), 4253-4264
MSC (1991): Primary 60G15, 60J65; Secondary 60J55
DOI: https://doi.org/10.1090/S0002-9947-98-02011-X
MathSciNet review: 1443196
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Abstract: The small ball problem for the integrated process of a real-valued Brownian motion is solved. In sharp contrast to more standard methods, our approach relies on the sample path properties of Brownian motion together with facts about local times and Lévy processes.


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  • 1. Albeverio, S., Hilbert, A. and Kolokoltsov, V.N. (1997). Transience of stochastically perturbed classical Hamiltonian systems and random wave operators. Stochastics Stochastics Rep. $\underline{60}$. CMP 97:09
  • 2. Albeverio, S., Hilbert, A. and Kolokoltsov, V.N. (1996). Estimates uniform in time for the transition probability of diffusions with small drift and for stochastically perturbed Newton equations. (preprint)
  • 3. Anderson, T.W. (1955). The integral of symmetric unimodular function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. $\underline{6}$ 170-176. MR 16:1005a
  • 4. P. Baldi and B. Roynette, Some exact equivalents for the Brownian motion in Hölder norm, Probab. Theory Related Fields 93 (1992), no. 4, 457–484. MR 1183887, https://doi.org/10.1007/BF01192717
  • 5. N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. MR 898871
  • 6. Chung, K.L. (1948). On the maximum partial sums of sequences of independent random variables. Trans. Amer. Math. Soc. $\underline{64}$ 205-233. MR 10:132b
  • 7. Deheuvels, P. and Mason, D.M. (1996). Random fractals and Chung-type functional laws of the iterated logarithm. (preprint).
  • 8. C. Donati-Martin and M. Yor, Fubini’s theorem for double Wiener integrals and the variance of the Brownian path, Ann. Inst. H. Poincaré Probab. Statist. 27 (1991), no. 2, 181–200 (English, with French summary). MR 1118933
  • 9. Bert Fristedt, Sample functions of stochastic processes with stationary, independent increments, Advances in probability and related topics, Vol. 3, Dekker, New York, 1974, pp. 241–396. MR 0400406
  • 10. M. Kac, Probability theory: Its role and its impact, SIAM Rev. 4 (1962), 1–11. MR 0151991, https://doi.org/10.1137/1004001
  • 11. Davar Khoshnevisan, Lévy classes and self-normalization, Electron. J. Probab. 1 (1996), no. 1, approx. 18 pp.}, issn=1083-6489, review=\MR{1386293},.
  • 12. A. A. Klyachko and Yu. V. Solodyannikov, Calculation of the characteristic functions of some functionals of the Wiener process and the Brownian bridge, Teor. Veroyatnost. i Primenen. 31 (1986), no. 3, 569–573 (Russian). MR 866878
  • 13. Frank B. Knight, Local variation of diffusion in local time, Ann. Probability 1 (1973), 1026–1034. MR 0397898
  • 14. Kolokoltsov, V.N. (1997). A note on the long time asymptotics of the Brownian motion with applications to the theory of quantum measurement. Potential Anal. $\underline{7}$ 759-764. CMP 98:04
  • 15. James Kuelbs and Wenbo V. Li, Small ball estimates for Brownian motion and the Brownian sheet, J. Theoret. Probab. 6 (1993), no. 3, 547–577. MR 1230346, https://doi.org/10.1007/BF01066717
  • 16. James Kuelbs and Wenbo V. Li, Metric entropy and the small ball problem for Gaussian measures, J. Funct. Anal. 116 (1993), no. 1, 133–157. MR 1237989, https://doi.org/10.1006/jfan.1993.1107
  • 17. J. Kuelbs, W. V. Li, and Qi Man Shao, Small ball probabilities for Gaussian processes with stationary increments under Hölder norms, J. Theoret. Probab. 8 (1995), no. 2, 361–386. MR 1325856, https://doi.org/10.1007/BF02212884
  • 18. James Kuelbs, Wenbo V. Li, and Michel Talagrand, liminf results for Gaussian samples and Chung’s functional LIL, Ann. Probab. 22 (1994), no. 4, 1879–1903. MR 1331209
  • 19. Lachal, A. (1992). Étude des Trajectoires de la Primitive du Mouvement Brownien. Thèse de Doctorat de l'Université Claude Bernard Lyon I.
  • 20. Aimé Lachal, Temps de sortie d’un intervalle borné pour l’intégrale du mouvement brownien, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 5, 559–564 (French, with English and French summaries). MR 1443994, https://doi.org/10.1016/S0764-4442(99)80390-5
  • 21. Wenbo V. Li, Lim inf results for the Wiener process and its increments under the 𝐿₂-norm, Probab. Theory Related Fields 92 (1992), no. 1, 69–90. MR 1156451, https://doi.org/10.1007/BF01205237
  • 22. Wenbo V. Li, Comparison results for the lower tail of Gaussian seminorms, J. Theoret. Probab. 5 (1992), no. 1, 1–31. MR 1144725, https://doi.org/10.1007/BF01046776
  • 23. Li, W.V. and Shao, Q.-M. (1996). Small ball estimates for Gaussian processes under Sobolev type norms. (preprint)
  • 24. H. P. McKean Jr., A winding problem for a resonator driven by a white noise, J. Math. Kyoto Univ. 2 (1963), 227–235. MR 0156389
  • 25. Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1994. MR 1303781
  • 26. Rice, S.O. (1944). Mathematical analysis of random noise. Bell Syst. Techn. J. $\underline{23}$ 282-332. MR 6:89b
  • 27. Q.-M. Shao and D. Wang, Small ball probabilities of Gaussian fields, Probab. Theory Related Fields 102 (1995), no. 4, 511–517. MR 1346263, https://doi.org/10.1007/BF01198847
  • 28. Z. Shi, Small ball probabilities for a Wiener process under weighted sup-norms, with an application to the supremum of Bessel local times, J. Theoret. Probab. 9 (1996), no. 4, 915–929. MR 1419869, https://doi.org/10.1007/BF02214257
  • 29. Stolz, W. (1995). Mesures Gaussiennes de Petites Boules et Petites Déviations. Thèse de Doctorat de l'Université Paul Sabatier Toulouse III.
  • 30. Wolfgang Stolz, Some small ball probabilities for Gaussian processes under nonuniform norms, J. Theoret. Probab. 9 (1996), no. 3, 613–630. MR 1400590, https://doi.org/10.1007/BF02214078
  • 31. Michel Talagrand, The small ball problem for the Brownian sheet, Ann. Probab. 22 (1994), no. 3, 1331–1354. MR 1303647
  • 32. H. F. Trotter, A property of Brownian motion paths, Illinois J. Math. 2 (1958), 425–433. MR 0096311
  • 33. Marc Yor, Some aspects of Brownian motion. Part I, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1992. Some special functionals. MR 1193919

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Additional Information

Davar Khoshnevisan
Affiliation: Department of Mathematics, Univeristy of Utah, Salt Lake City, Utah 82112
Email: davar@math.utah.edu

Zhan Shi
Affiliation: L.S.T.A., Université Paris VI, 4, Place Jussieu, 75252 Paris Cedex 05, France
Email: shi@ccr.jussieu.fr

DOI: https://doi.org/10.1090/S0002-9947-98-02011-X
Keywords: Small ball probability, integrated Brownian motion
Received by editor(s): October 19, 1996
Received by editor(s) in revised form: January 3, 1997
Additional Notes: Research partially supported by grants from the National Science Foundation and the National Security Agency
Article copyright: © Copyright 1998 American Mathematical Society