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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 100, Issue 3
  • November 1986, pp. 383-406

The measure theory of random fractals

  • S. James Taylor (a1)
  • DOI:
  • Published online: 24 October 2008

In 1951 A. S. Besicovitch, who was my research supervisor, suggested that I look at the problem of determining the dimension of the range of a Brownian motion path. This problem had been communicated to him by C. Loewner, but it was a natural question which had already attracted the attention of Paul Lévy. It was a good problem to give to an ignorant Ph.D. student because it forced him to learn the potential theory of Frostman [33] and Riesz[75] as well as the Wiener [98] definition of mathematical Brownian motion. In fact the solution of that first problem in [81] used only ideas which were already twenty-five years old, though at the time they seemed both new and original to me. My purpose in this paper is to try to trace the development of these techniques as they have been exploited by many authors and used in diverse situations since 1953. As we do this in the limited space available it will be impossible to even outline all aspects of the development, so I make no apology for giving a biased account concentrating on those areas of most interest to me. At the same time I will make conjectures and suggest some problems which are natural and accessible in the hope of stimulating further research.

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[2]R. J. Adler . The uniform dimension of the level sets of a Brownian sheet. Ann. Probab. 6 (1978), 509515.

[7]R. M. Blumenthal and R. K. Getoor . Some theorems on stable processes. Trans. Amer. Math. Soc. 95 (1960), 263273.

[12]Z. Ciesielski and S. J. Taylor . 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 (1962), 434450.

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[15]P. L. Davies . The exact Hausdorff measure of the zero set of certain stationary Gaussian processes. Ann. Probab. 5 (1977), 740755.

[17]B. Davis . On the paths of symmetric stable processes. Z. Wahrscheinlichkeitstheorie 64 (1983), 359367.

[18]A. Dvoretzky . On the oscillation of the Brownian motion process. Israel J. Math. 1 (1963), 212214.

[24]E. B. Dynkin Random fields associated with multiple points of Brownian motion. Jour. Functional Anal. 62 (1985), 397434.

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[30]B. E. Fristedt . An extension of a theorem of S. J. Taylor concerning multiple points of the symmetric stable process. Z. Wahrscheinlichkeitstheorie 9 (1967), 6264.

[31]B. E. Fristedt and W. E. Pruitt . Lower functions for increasing random walks and subordinators. Z. Wahrscheinlichkeitstheorie 18 (1971), 167182.

[34]M. Fukushima . Basic properties of Brownian motion and a capacity on the Weiner space. J. Math. Soc. Japan36 (1984), 161176.

[35]D. Geman and J. Horowitz . Occupation densities. Ann. Probab. 8 (1980), 167.

[36]P. Greenwood and E. A. Perkins . A conditioned limit theorem for random walk and Brownian local time on square root boundaries. Ann. Probab. 11 (1983), 227261.

[37]J. Hawkes . On the Hausdorff dimension of the range of a stable process with a Borel set. Z. Wahrscheinlichkeitstheorie 19 (1971), 90102.

[38]J. Hawkes . The measure of the range of a subordinator. Bull. London Math. Soc. 5 (1973), 2128.

[39]J. Hawkes . Random recorderings of intervals complementary to a linear set. Quart. J. Math. Oxford35 (1984), 165172.

[40]J. Hawkes and W. E. Pruitt . Uniform dimension results for processes with independent increments. Z. Wahrscheinlichkeitstheorie 28, 277288.

[41]W. J. Hendricks . Hausdorff dimension in a process with stable components—an interesting counterexample. Ann. Math. Statis. 43 (1972), 690694.

[42]W. J. Hendricks . Multiple points for a process in with stable components. Z. Wahrscheinlichkeitstheorie 28 (1974), 113128.

[43]W. J. Hendricks . Multiple points for transient symmetric Lévy processes in . Z. Wahrscheinlichkeitstheorie 49 (1979), 1321.

[45]N. Jain and W. E. Pruitt . The correct measure function for the graph of a transient stable process. Z. Wahrscheinlichkeitstheorie 9 (1968), 131138.

[52]W. S. Kendall . Contours of Brownian processes with several dimensional time. Z. Wahrscheinlichkeitstheorie 52 (1980), 267276.

[55]N. Kono . The exact Hausdorff measure of irregularity points for a Brownian path. Z. Wahrscheinlichkeitstheorie 40 (1977), 257282.

[62]H. P. McKean . Hausdorff-Besicovitch dimension of Brownian motion paths. Duke J. 22 (1955), 229234.

[63]H. P. McKean . Sample functions of stable processes. Ann. Math. 61 (1955), 564579.

[64]S. Orey . Gaussian sample functions and Hausdorff dimension of level crossings. Z. Wahrscheinlichkeitstheorie 15 (1970), 249256.

[66]E. A. Perkins . The exact Hausdorff measure of the level sets of Brownian motion. Z. Wahrscheinlichkeitstheorie 58 (1981), 373388.

[67]E. A. Perkins . On the Hausdorff dimension of Brownian slow points. Z. Wahrscheinlichkeitstheorie 64 (1983), 369399.

[70]L. Pitt . Local times for Gaussian vector fields. Indiana Univ. Math. J. 27 (1978), 309330.

[72]W. E. Pruitt and S. J. Taylor . Sample path properties of processes with stable components. Z. Wahrscheinlichkeitstheorie 12 (1969), 267289.

[73]W. E. Pruitt and S. J. Taylor . Hausdorff measure properties of the asymmetric Cauchy processes. Ann. Probab. 5 (1977), 608615.

[74]D. Ray . Sojourn times and the exact Hausdorff measure of the sample path for planar Brownian motion. Trans. Amer. Math. Soc. 106 (1963), 436444.

[77]J. Rosen . Self-intersections of random fields. Ann. Probab. 12 (1984), 108119.

[85]S. J. Taylor . Multiple points for the sample paths of the symmetric stable process. Z. Wahrscheinlichkeitstheorie 5 (1966), 247264.

[89]S. J. Taylor and C. Tricot . Packing measure, and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288 (1985), 679699.

[91]S. J. Taylor and J. G. Wendel . The exact Hausdorff measure of the zero set of a stable process. Z. Wahrscheinlichkeitstheorie 6 (1966), 170180.

[94]L. T. Tran . The Hausdorff dimension of the range of the N-parameter Wiener process. Ann. Probab. 5 (1977), 235242.

[98]N. Wiener . Differential space. J. Math. Phys. 2 (1923), 131174.

[99]R. L. Wolpert . Wiener path intersections and local time. J. Functional Anal. 30 (1978), 329340.

[100]L. Yoder . The Hausdorff dimensions of the graph and range of the N-parameter Brownian motion in d-space. Ann. Probab. 3 (1975), 169171.

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