Summary
LetX t be a Brownian motion and letS(c) be the set of realsr≧0 such that üX r+t −X r ü≦c√t, 0≦t≦h, for someh=h(r)>0. It is known thatS(c) is empty ifc<1 and nonempty ifc>1, a.s. In this paper we prove thatS(1) is empty a.s.
References
[BP] M.T. Barlow, E.A. Perkins: Brownian motion at a slow point. Trans. Amer. Math. Soc.296, 741–775 (1986)
[CH] R. Courant, D. Hilbert: Methods of mathematical physics, Vol. 1. Interscience: New York, 1953
[Da] B. Davis: On Brownian slow points. Z. Wahrsch. Verw. Geb.64, 359–367 (1983)
[DP] B. Davis, E.A. Perkins: Brownian slow points: the critical case. Ann. Probab.13, 779–803 (1985)
[D] A. Dvoretzky: On the oscillation of the Brownian motion process. Isr. J. Math.1, 212–214 (1963)
[GP] P. Greenwood, E.A. Perkins: A conditioned limit theorems for random walk and Brownian local time on square root boundaries. Ann. Probab.11, 227–261 (1983)
[K1] J.-P. Kahane: Sur l'irregularité locale du mouvement Brownien. C.R. Acad. Sci. Paris278, 331–333 (1974)
[K2] J.-P. Kahane: Sur les zéros et les instants de relantissement du mouvement Brownien. C.R. Acad. Sci. Paris282, 431–433 (1976)
[Kn] F.B. Knight: Essentials of Brownian motion and diffusion. Amer. Math. Soc.: Providence, 1981
[N] A.A. Novikov: On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary. Math. USSR Sbornik38, 495–505 (1981)
[P] E.A. Perkins: On the Hausdorff dimension of the Brownian slow points. Z. Wahrsch. Verw. Geb.64, 369–399 (1983)
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This research was partially supported by NSF Grant 9322689.
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Bass, R.F., Burdzy, K. A critical case for Brownian slow points. Probab. Th. Rel. Fields 105, 85–108 (1996). https://doi.org/10.1007/BF01192072
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DOI: https://doi.org/10.1007/BF01192072
Mathematics Subject Classification (1991)
- 60G17
- 60J65