Abstract
We study the almost sure asymptotic behaviors of the Lebesgue measure of the points which are hardly visited, in the sense of Földes and Révész,(7) by a linear Wiener process.
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REFERENCES
Bass, R. F., and Griffin, P. S. (1985). The most visited site of Brownian motion and simple random walk. Z. Wahrsch. verw. Gebiete 70, 417–436.
Borodin, A. N., and Salminen, P. (1996). Handbook of Brownian Motion—Facts and Formulae, Birkhäuser, Boston.
Ciesielski, Z., and Taylor, S. J. (1962). 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, 434–450.
Eisenbaum, N. (1989). Temps locaux, excursions et lieu le plus visité par un mouvement Brownien linéaire. Thèse de Doctorat de l'Université Paris VII, 1989.
Eisenbaum, N. (1997). On the most visited sites by a symmetric stable process. Prob. Th. Rel. Fields 107, 527–535.
Földes, A. (1989). On the infimum of the local time of a Wiener process. Prob. Th. Rel. Fields 82, 545–563.
Földes, A., and Révész, P. (1992). On hardly visited points of the Brownian motion. Prob. Th. Rel. Fields 91, 71–80.
Gruet, J.-C., and Shi, Z. (1996). The occupation time of Brownian motion in a ball. J. Theoret. Prob. 9, 429–445.
Kent, J. (1978). Some probabilistic properties of Bessel functions. Ann. Prob. 6, 760–770.
Kiefer, J. (1959). K-sample analogues of the Kolmogorov-Smirnov and Cramér-v. Mises tests. Ann. Math. Statist. 30, 420–447.
Kochen, S. B., and Stone, C. J. (1964). A note on the Borel-Cantelli lemma. Illinois J. Math. 8, 248–251.
Leuridan, C. (1997). Le point d'un fermé le plus visité par le mouvement Brownien. Ann. Prob. 25, 953–996.
Pitman, J. W., and Yor, M. (1996). Decomposition at the maximum for excursions and bridges of one dimensional diffusions. In Ikeda, N., Watanabe, S., Fukushima, M., and Kunita, H. (eds.), Itô's Stochastic Calculus and Probability Theory, Springer, Tokyo, pp. 293–310.
Révész, P. (1990). Random Walk in Random and Non-Random Environments, World Scientific, Singapore.
Revuz, D., and Yor, M. (1999). Continuous Martingales and Brownian Motion, Third Edition, Springer, Berlin.
Trotter, H. F. (1958). A property of Brownian motion paths. Illinois J. Math. 2, 425–433.
Yor, M. (1995). Local Times and Excursions for Brownian Motion: A Concise Introduction. Lecciones en Matemáticas, Universidad Central de Venezuela.
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Eisenbaum, N., Shi, Z. Measuring the Rarely Visited Sites of Brownian Motion. Journal of Theoretical Probability 12, 595–613 (1999). https://doi.org/10.1023/A:1021615513008
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DOI: https://doi.org/10.1023/A:1021615513008