On the sample paths of the symmetric stable processes in spaces

Junji TAKEUCHI

doi:10.2969/jmsj/01620109

Junji TAKEUCHI

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JOURNAL FREE ACCESS

1964 Volume 16 Issue 2 Pages 109-127

DOI https://doi.org/10.2969/jmsj/01620109

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Published: 1964 Received: September 02, 1963 Available on J-STAGE: September 26, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: September 26, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) A. S. Besicovitch, On the existence of subsets of finite measure of sets of infinite measure, Indag. Math., 14 (1952), 339-344.
2) R. M. Blumenthal and R. K. Getoor, A dimension theorem for sample functions of stable processes, Illinois J. Math., 4 (1960), 370-375.
3) R. M. Blumenthal and R. K. Getoor, The dimension of the set of zeros and the graph of a symmetric stable process, Illinois J. Math., 6 (1962), 308-316.
4) K.L. Chung and P. Erdös, On the application of the Borel-Cantelli lemma, Trans. Amer. Math. Soc., 72 (1952), 179-186.
5) J.L. Doob, Semimartingales and subharmonic functions, Trans. Amer. Math. Soc., 77 (1954), 86-121.
6) A. Dvoretzky and P. Erdös, Some problems on random walk in space, Proc. Second Berkeley Symposium of Mathematical Statistics and Probability, 1951, 353-367.
7) A. Dvoretzky, P. Erdös and S. Kakutani, Double points of paths of Brownian motion in n-space, Acta Sci. Math. Szeged., 12 (1950), 75-81.
8) K. Ito, Lectures on stochastic processes, Tata Institute of Fundamental Research, Bombay, 1961.
9) S. Kametani, On Hausdorff's measures and generalized capacities with some of their applications to the theory of functions, Japan. J. Math., 19 (1945), 217-257.
10) A. Khintchine, Zwei Sätze über stochastische Prozesse mit stabilen Verteilun-gen, Mat. Sbornik, 45 (1938), 577-584, (Russian, German summary).
11) P. Lévy, Le mouvement Brownien plan, Amer. J. Math., 62 (1940), 487-550.
12) H. P. McKean, Sample functions of stable processes, Ann. of Math., 61 (1955), 564-579.
13) M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math. Szeged., 9 (1938), 1-42.
14) S. Saks, Theory of the integral, Warszawa-Lwów, 1937.
15) F. Spitzer, Some theorems concerning 2-dimensional Brownian motion, Trans. Amer. Math. Soc., 87 (1958), 187-197.
16) J. Takeuchi, T. Yamada and S. Watanabe, Stable process - Riesz potential, properties of the paths-, Seminar on Prob., 13, 1962, pp. 119, (Japanese).
17) S. J. Taylor, On the connection between Hausdorff measures and generalized capacities, Proc. Cambridge Philos. Soc., 57 (1961), 524-531.
18) S. Watanabe, On stable processes with boundary conditions, J. Math. Soc. Japan, 14 (1962), 170-198.

Right : [1] A. S. Besicovitch, On the existence of subsets of finite measure of sets of infinite measure, Indag. Math., 14 (1952), 339-344.
[2] R. M. Blumenthal and R. K. Getoor, A dimension theorem for sample functions of stable processes, Illinois J. Math., 4 (1960), 370-375.
[3] R. M. Blumenthal and R. K. Getoor, The dimension of the set of zeros and the graph of a symmetric stable process, Illinois J. Math., 6 (1962), 308-316.
[4] K. L. Chung and P. Erdös, On the application of the Borel-Cantelli lemma, Trans. Amer. Math. Soc., 72 (1952), 179-186.
[5] J. L. Doob, Semimartingales and subharmonic functions, Trans. Amer. Math. Soc., 77 (1954), 86-121.
[6] A. Dvoretzky and P. Erdös, Some problems on random walk in space, Proc. Second Berkeley Symposium of Mathematical Statistics and Probability, 1951, 353-367.
[7] A. Dvoretzky, P. Erdös and S. Kakutani, Double points of paths of Brownian motion in n-space, Acta Sci. Math. Szeged., 12 (1950), 75-81.
[8] K. Itô, Lectures on stochastic processes, Tata Institute of Fundamental Research, Bombay, 1961.
[9] S. Kametani, On Hausdorff's measures and generalized capacities with some of their applications to the theory of functions, Japan. J. Math., 19 (1945), 217-257.
[10] A. Khintchine, Zwei Sätze über stochastische Prozesse mit stabilen Verteilungen, Mat. Sbornik, 45 (1938), 577-584, (Russian, German summary).
[11] P. Lévy, Le mouvement Brownien plan, Amer. J. Math., 62 (1940), 487-550.
[12] H. P. McKean, Sample functions of stable processes, Ann. of Math., 61 (1955), 564-579.
[13] M. Riesz, Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math. Szeged., 9 (1938), 1-42.
[14] S. Saks, Theory of the integral, Warszawa-Lwów, 1937.
[15] F. Spitzer, Some theorems concerning 2-dimensional Brownian motion, Trans. Amer. Math. Soc., 87 (1958), 187-197.
[16] J. Takeuchi, T. Yamada and S. Watanabe, Stable process -Riesz potential, properties of the paths-, Seminar on Prob., 13, 1962, pp. 119, (Japanese).
[17] S. J. Taylor, On the connection between Hausdorff measures and generalized capacities, Proc. Cambridge Philos. Soc., 57 (1961), 524-531.
[18] S. Watanabe, On stable processes with boundary conditions, J. Math. Soc. Japan, 14 (1962), 170-198.

Date of correction: September 26, 2006 Reason for correction: - Correction: PDF FILE Details: -

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