Construction and multifractal analysis of random functions and their graphs

Abstract : This thesis deals with the construction and multifractal analysis of random functions and their graphs. At first, we contribute to Kahane's T-martingale theory by considering complex [0, 1]- martingales. While until now this is done with positive [0; 1]-martingales, in particular in order to build singular measures with respect to the Lebesgue, we construct complex continuous function-valued martingales and consider the question of their almost sure uniform convergence. We get a general sufficient condition for such a convergence to hold for the elements of a large subclass of [0, 1]-martingales. All the non-degenerate limit functions are candidates to be multifractal. Their multifractal analysis reveals new difficulties. We conduct this multifractal analysis for complex "b-adic independent cascade functions". This study leads to new interesting phenomena. In particular, we build statistically self-similar continuous functions whose singularity spectrum is left-sided and supported by the whole interval [0;1]. Further, we consider new singularity spectra associated with the graph, range and level sets of multifractal functions. These spectra consist in calculating the Hausdorff dimension of the iso-Hölder sets put on the graph, range and level sets. For real-valued b-adic independent cascade functions, with probability 1, we obtain the graph and range singularity spectra, as well as the level set singularity spectrum in "Lebesgue almost every directions", for the so-called non-conservative b-adic independent cascade functions. Finally, we consider the same question for another class of random multifractal functions, namely random wavelet series built from Gibbs measures. Under suitable assumptions, we obtain the graph and range singularity spectra almost surely.
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Xiong Jin. Construction and multifractal analysis of random functions and their graphs. Classical Analysis and ODEs [math.CA]. Université Paris Sud - Paris XI, 2010. English. ⟨tel-00841501⟩

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