Stochastic Analysis of Random Multifractal Measures

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Fractals are defined by the repetition or near-repetition of a pattern on infinitely many scales. They appear extensively in nature and are prevalent in human-generated data such as financial time series, sound files, and internet traffic patterns. By their very nature fractals are rough, non-differentiable objects that cannot be studied by the classical mathematical methods of calculus. The more modern tools of multifractal analysis, which quantify the degree of roughness that appears on various scales, can be used instead. It is particularly interesting to apply multifractal analysis to randomly generated objects, which by default tend to exhibit fractal behavior. Random measures, which for example describe the rainfall distribution over a particular state or the distribution of energy in turbulent fluid flows, are a particularly fruitful area for employing multifractal analysis. The results of such an analysis can often be used for prediction and forecasting of the underlying random system. This research project aims to further develop the mathematical foundations for analysis of multifractal properties.The aim of this project is to analyze the multifractal properties of a broad class of random measures that appear in models of statistical mechanics. The focus will be on random pinning models, random polymer models, random measures arising in two-dimensional conformally invariant systems, and spectral measures of random matrices. In all cases, the major quantity of interest will be the multifractal spectrum, which quantifies the amount of mass the random measure assigns around various points in the space. Computation of the multifractal spectrum is closely related to the theory of large deviations for computing the decay of probabilities of rare events. Many of the tools from that theory will be employed in this project, including the notion of Gibbs condition for describing the most likely behavior of the random system when a rare event occurs. As all the random measures listed above can be described using tools from stochastic analysis, specifically the Wiener chaos decomposition, a particular goal of this project will be to incorporate tools from stochastic analysis into this theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.