Abstract
The Renyi function for the logical time measure μ of Brownian motion is found. It is shown that this function, the Legendre transform of the multifractal spectrum of μ and the τ-function derived by the reciprocal measure formalism are not identical. More examples of μ having similar anomalies are discussed.
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REFERENCES
J. Bertoin, Lévy processes, in Cambridge Tracts in Mathematics, Vol. 121 (Cambridge Univ. Press, 1996).
J. Bertoin, The inviscid Burgers equation with Brownian initial velocity, Preprint (1997).
R. Cawley and R. D. Mauldin, Multifractal decomposition of Moran fractals, Adv. Math. 92:196–236 (1992).
P. Collet and F. Koukiou, Large deviations for multiplicative chaos, Commun. Math. Phys. 147:329–342 (1992).
D. Dolgopyat and V. Sidorov, Multifractal properties of the sets of zeroes of Brownian paths, Fund. Math. 147:2, 157–171 (1995).
K. J. Falconer, The multifractal spectrum of statistically self-similar measures, J. Theoret. Prob. 7:3, 681–702 (1996).
X. Fernique, Regularite des trajectories des functions aleatoires gaussiennes, Lectures Notes in Mathematics, Vol. 480:2–187 (1975).
U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge Univ. Press, 1995).
D. Geman and J. Horowitz, Occupation densities, Ann. Probab. 8:1–67 (1980).
T. C. Hulsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. J. Shraiman, Fractal measures and their singularities: The characterization of strange sets, Phys. Rev. A 33:1141–1151 (1986).
K. Ito and H. P. McKean, Jr., Diffusion Processes and Their Sample Paths (Springer-Verlag, 1965).
S. Jaffard, The multifractal nature of Levy processes, Preprint (1997).
J. P. Kahane, Some random series of functions, in Cambridge Studies in Advanced Mathematics, Vol. 5 (Cambridge Univ. Press, 1985), p. 305.
R. D. Mauldin and M. Urbański, Dimensions and measure in infinite iterated function systems, Proc. London. Math. Soc. (3) 73:105–154 (1996).
G. M. Molchan, Multifractal analysis of Brownian zero set, J. Stat. Phys. 79(3/4):701–730 (1995).
G. M. Molchan, Scaling exponents and multifractal dimensions for independent random cascades, Commun. Math. Phys. 179:681–702 (1996).
G. M. Molchan, Turbulent cascades: Limitations and statistical test of the log-normal hypothesis, Physics of Fluids 9:(8), 2387–2396 (1997).
G. Molchan and Ju. Golosov, Gaussian stationary processes with asymptotic power spectrum, Soviet. Math. Dokl. 10(1):134–137 (1969).
E. Nummelin, General irreducible Markov chains and non-negative operators (Cambridge Univ. Press, 1984).
L. Olsen, Random Geometrically Graph Directed Self-similar Multifractals, Pitman Research Notes in Math. Ser. 307. (Longman, Harlow, 1994).
L. Olsen, A Multifractal formalism, Adv. Math. 116:82–195 (1995).
S. Orey and S. J. Taylor, How often on a Brownian path does the law of the iterated logarithm fail? Proc. London Math. Soc. (3) 28:174–192 (1974).
G. Parisi and U. Frisch, On the singularity structure of fully developed turbulence, in Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, M. Ghil, R. Benzi, and G. Parisi, eds. (North-Holland, Amsterdam, 1985), pp. 84–88.
R. H. Reid and B. Mandelbrot, Multifractal formalism for multinomial measures, Adv. in Appl. Math. 16:132–150 (1995).
Z. She, E. Aurell, and U. Frisch, The inviscid Burgers equation with initial data of Brownian type, Commun. Math. Phys. 148:623–641 (1992).
Y. G. Sinai, Statistics of shocks in solutions of the inviscid Burgers equation, Commun. Math. Phys. 148:601–621 (1992).
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Molchan, G.M. Anomalies in Multifractal Formalism for Local Time of Brownian Motion. Journal of Statistical Physics 91, 199–220 (1998). https://doi.org/10.1023/A:1023044221942
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DOI: https://doi.org/10.1023/A:1023044221942