Summary
It is shown that, for a Wiener process X t , both the quantities \(\mathop {\inf }\limits_t \mathop {\overline {\lim } }\limits_{h{\text{ }} \to {\text{ 0 + }}} |X_{t{\text{ }} + {\text{ }}h} - X_t |/\sqrt h {\text{ and }}\mathop {{\text{sup}}}\limits_t \mathop {\underline {\lim } }\limits_{h{\text{ }} \to {\text{ 0 + }}} {\text{ }}(X_{t{\text{ }} + {\text{ }}h} - X_t )/\sqrt h \) are almost surely equal to 1.
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Dvoretzky, A.: On the oscillation of the Brownian motion process. Israel J. Math. 1, 212–214 (1963)
Kahane, J.-P.: Sur l'irrégularité locale du mouvement brownien. C.R. Acad. Sci. Paris 278, 331–333 (1974)
Kahane, J.-P.: Sur les zéros et les instants de ralentissement du mouvement brownien. C.R. Acad. Sci Paris 282, 431–433 (1976)
Kahane, J.-P.: Slow points of Gaussian processes. Preprint.
Knight, F.B.: Essentials of Brownian motion and diffusion. Amer. Math. Soc. Math. Surveys; no. 18 (1981).
Novikov, A.A.: On moment inequalities for stochastic integrals. Theory Probability Appl. (English translation) 16, 449–456 (1971)
Paley, R.E.A.C., Wiener, N., Zygmund, A.: Notes on random functions. Math. Z. 37, 647–668 (1932)
Shepp, L.A.: A first passage problem for the Wiener process. Ann. Math. Statist. 38, 1912–1914 (1967)
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Davis, B. On Brownian slow points. Z. Wahrscheinlichkeitstheorie verw Gebiete 64, 359–367 (1983). https://doi.org/10.1007/BF00532967
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DOI: https://doi.org/10.1007/BF00532967