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Reports (Research Report) Year : 2002

Multiperiodic multifractal martingale measures

Julien Barral
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  • IdRef : 14587494X
Marc-Olivier Coppens
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Benoît B. Mandelbrot
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Abstract

A nonnegative 1-periodic multifractal measure on R is obtained as infinite random product of harmonics of a 1-periodic function W(t). Such infinite products are statistically self-affine and generalize certain Riesz products with random phases. This convergence is due to their martingale structure. The criterion of non-degeneracy is provided. It differs from those of other random measures constructed as martingale limits of multiplicative processes. It is also very sensitive to small changes in W(t). Interpreting these infinite products in the framework of thermodynamic formalism for random transformations makes these infinite product non-degenerate and convergent via a natural normalization that does not affect non-degenerate original infinite products. The multifractal analysis of the limit measure is studied. It requires suitable Gibbs measures. In the thermodynamic formalism, the notion of weak Gibbs measures was recently introduced and it is associated with a weak principle of bounded variations for the potential function. There, the potential belongs to a subclass of piecewise continuous functions; here, the role of the potential is played by the logarithm of W. A new approach we develop makes it possible to obtain the multifractal nature of infinite random products of harmonics of periodic functions W with a dense countable set of jump points.
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Dates and versions

inria-00072025 , version 1 (23-05-2006)

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  • HAL Id : inria-00072025 , version 1

Cite

Julien Barral, Marc-Olivier Coppens, Benoît B. Mandelbrot. Multiperiodic multifractal martingale measures. [Research Report] RR-4563, INRIA. 2002. ⟨inria-00072025⟩
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