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Multifractal analysis of path regularity of Lévy-type processes

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FD2W02 - Fractional kinetics, hydrodynamic limits and fractals

Let $K$ be a closed subset of $\mathbb{R}d$. The $d-$dimensional full $K-$moment problem asks when a linear functional on the algebra of polynomials in $d$ variables, that is a sequence of putative moments, has an integral representation w.r.t. a measure supported in $K$. The same question can be posed for a general algebra $A$ (not necessarily the polynomial one) where the linear space $\mathbb{R}d$ is replaced by the space of characters of $A$. In this talk, I will present recent results on the infinite dimensional full $K-$moment problem, that is either $K$ is an infinite dimensional closed set or $A$ cannot be generated by finitely many elements. I will give an overview of recently achieved results on this topic and describe how projective limit techniques can shed a light on the fundamental differences between the finite and the infinite dimensional moment problem.As for the case $K-$compact, which has been heightening the interest in the moment problem since the seminal work of K.~Schm\”udgen in 1991, I will present an exact characterization of the compact support of the representing measure. In fact, all previous results only provide representing measure with compact support contained in $K$ and typically strictly smaller, while we can provide exact descriptions of the support.Finally, I give some results on the moment problem for random measures and point processes, which significantly improve known necessary and sufficient conditions for the existence of a representing measure in those cases. These demonstrates the power of the above described techniques to systematically take into account prescribed structures. 

This talk is part of the Isaac Newton Institute Seminar Series series.

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