@MISC{Jaffard_themultifractal, author = {Stéphane Jaffard}, title = {The multifractal nature of Lévy processes}, year = {} }

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. We show that the sample paths of most L'evy processes are multifractal functions and we determine their spectrum of singularities. Key Words. L'evy processes, multifractals, Holder singularities, Hausdorff dimensions, spectrum of singularities. AMS Classification. 28A80, 60G17, 60G30, 60J30, A L'evy process X t (t 0) valued in IR d is, by definition, a stochastic process with stationary independent increments: X t+s \Gamma X t is independent of the (X v ) 0vt and has the same law as X s . Brownian motion and Poisson processes are examples of L'evy processes that can be qualified as monofractal; for instance the Holder exponent of the Brownian motion is everywhere 1=2 (the variations of its regularity are only of a logarithmic order of magnitude). These two examples are not typical: we will see that the other L'evy processes are multifractal provided that their L'evy measure is neither too small nor too large near zero. Furthermore their spectrum of singularities depends precise...