On fractal minimizers and potentials of occupation measures

TL;DR

This paper demonstrates the existence of fractal minimizers in variational problems, showing their properties resemble stochastic processes and establishing new results for potentials of occupation measures of Gaussian fields.

Contribution

It introduces novel existence proofs for fractal minimizers and extends continuity results for occupation measure potentials of Gaussian fields.

Findings

Existence of fractal minimizers with non-integer Hausdorff dimension

New continuity and boundedness results for occupation measure potentials

Generalization of fractional Sobolev and BV function compositions

Abstract

We consider four prototypes of variational problems and prove the existence of fractal minimizers through the direct method in the calculus of variations. By design these minimizers are H\"older curves or H\"older parametrizations of hypersurfaces whose images generally have a non-integer Hausdorff dimension. Although their origin is deterministic, their regularity properties are roughly similar to those of typical realizations of stochastic processes. As a key tool, we prove novel continuity and boundedness results for potentials of occupation measures of Gaussian random fields. These results complement well-known results for local times, but hold under much less restrictive assumptions. In an auxiliary section, we generalize earlier results on non-linear compositions of fractional Sobolev functions with $BV$-functions to higher dimensions.