Mathieu Helfter (ISTA)
In this talk, I will present a framework for multifractal analysis of infinite-dimensional measures. This us based on the notion of “scales,” which are finite metric invariants that generalize classical notions such as Hausdorff dimension and pointwise dimension of measures to both finite and infinite-dimensional spaces. We will focus in particular on the paradigmatic example of Brownian motion, whose multifractal spectrum can be described using measures associated with fractional Brownian motions. Since the Wiener measure cannot serve as a Haar measure, it is necessarily concentrated on a rich set of non-typical trajectories, whose scaling irregularities reveal its multifractal nature. We will also discuss several new questions and problems arising in this framework. The talk is based on joint work in progress with Aihua Fan.