First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path

Ciesielski, Z.; Taylor, S. J.

doi:10.1090/S0002-9947-1962-0143257-8

First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path

Authors: Z. Ciesielski and S. J. Taylor
Journal: Trans. Amer. Math. Soc. 103 (1962), 434-450
MSC: Primary 60.62
DOI: https://doi.org/10.1090/S0002-9947-1962-0143257-8
MathSciNet review: 0143257
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R. Courant and D. Hilbert, Methoden der Mathematischen Physik, Vols. 1, 2, Springer, Berlin, 1937.

D. A. Darling and M. Kac, On occupation times for Markoff processes, Trans. Amer. Math. Soc. 84 (1957), 444–458. MR 84222, DOI https://doi.org/10.1090/S0002-9947-1957-0084222-7
J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Limited, London, 1953. MR 0058896
A. Dvoretzky and P. Erdös, Some problems on random walk in space, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950., University of California Press, Berkeley and Los Angeles, 1951, pp. 353–367. MR 0047272
Paul Erdös, On the law of the iterated logarithm, Ann. of Math. (2) 43 (1942), 419–436. MR 6630, DOI https://doi.org/10.2307/1968801
P. Erdős and S. J. Taylor, Some problems concerning the structure of random walk paths, Acta Math. Acad. Sci. Hungar. 11 (1960), 137–162. (unbound insert) (English, with Russian summary). MR 121870, DOI https://doi.org/10.1007/BF02020631
P. Erdős and S. J. Taylor, On the Hausdorff measure of Brownian paths in the plane, Proc. Cambridge Philos. Soc. 57 (1961), 209–222. MR 126892, DOI https://doi.org/10.1017/s0305004100035076

F. Hausdorff, Dimension und äusseres Mass, Math. Ann. 79 (1919), 157-179.

I. I. Hirschman and D. V. Widder, The convolution transform, Princeton University Press, Princeton, N. J., 1955. MR 0073746
M. Kac, On some connections between probability theory and differential and integral equations, Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles, 1951, pp. 189–215. MR 0045333
J. Kiefer, $K$-sample analogues of the Kolmogorov-Smirnov and Cramér-V. Mises tests, Ann. Math. Statist. 30 (1959), 420–447. MR 102882, DOI https://doi.org/10.1214/aoms/1177706261
Paul Lévy, La mesure de Hausdorff de la courbe du mouvement brownien, Giorn. Ist. Ital. Attuari 16 (1953), 1–37 (1954) (French). MR 64344
C. A. Rogers and S. J. Taylor, The analysis of additive set functions in Euclidean space, Acta Math. 101 (1959), 273–302. MR 107690, DOI https://doi.org/10.1007/BF02559557
C. A. Rogers and S. J. Taylor, Functions continuous and singular with respect to a Hausdorff measure, Mathematika 8 (1961), 1–31. MR 130336, DOI https://doi.org/10.1112/S0025579300002084
M. Rosenblatt, On a class of Markov processes, Trans. Amer. Math. Soc. 71 (1951), 120–135. MR 43406, DOI https://doi.org/10.1090/S0002-9947-1951-0043406-9
S. J. Taylor, The Hausdorff $\alpha $-dimensional measure of Brownian paths in $n$-space, Proc. Cambridge Philos. Soc. 49 (1953), 31–39. MR 52719, DOI https://doi.org/10.1017/s0305004100028000
S. J. Taylor, On the connexion between Hausdorff measures and generalized capacity, Proc. Cambridge Philos. Soc. 57 (1961), 524–531. MR 133420, DOI https://doi.org/10.1017/s0305004100035581
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746