Optimal transport for multifractal random measures. Applications

TL;DR

This paper develops a new multistep optimal transport framework for irregular multifractal random measures, enabling applications in constructing multifractal time changes and random metrics related to KPZ problems.

Contribution

It introduces a novel transportation notion suited for irregular measures and demonstrates its applications in multifractal analysis and random geometry.

Findings

New multistep transportation concept for irregular measures

Construction of multifractal time changes

Existence of random metrics matching multifractal measures

Abstract

In this paper, we study optimal transportation problems for multifractal random measures. Since these measures are much less regular than optimal transportation theory requires, we introduce a new notion of transportation which is intuitively some kind of multistep transportation. Applications are given for construction of multifractal random changes of times and to the existence of random metrics, the volume forms of which coincide with the multifractal random measures. This study is motivated by recent problems in the KPZ context.