Flows driven by multi-indices Rough Paths

TL;DR

This paper develops a solution theory for scalar rough differential equations driven by multi-indices rough paths, utilizing the flow approach and log-ODE method, and explores the effects of translating multi-indices rough paths.

Contribution

It introduces a novel solution framework for multi-indices rough paths using the flow approach and extends the understanding of translation effects in this context.

Findings

Established a solution theory for multi-indices rough paths

Demonstrated the compatibility of the flow approach with this setting

Analyzed the translation action on rough differential equations

Abstract

In this work, we introduce a solution theory for scalar-valued rough differential equations driven by multi-indices rough paths. To achieve this task, we will show how the flow approach using the log-ODE method introduced by Bailleul fits perfectly in this setting. In addition, we also describe the action of the translation of multi-indices rough paths at the level of rough differential equations.