Unbounded rough drivers

TL;DR

This paper develops a theory for solving linear differential equations driven by unbounded operator-valued rough signals, with applications to rough linear transport and hyperbolic systems driven by distributional vector fields.

Contribution

It introduces a novel framework for analyzing differential equations driven by unbounded rough signals, extending rough path theory to new classes of equations.

Findings

Established a theory for unbounded operator-valued rough signals

Applied the theory to rough linear transport equations

Extended analysis to hyperbolic systems with distributional drivers

Abstract

We propose a theory of linear differential equations driven by unbounded operator-valued rough signals. As an application we consider rough linear transport equations and more general linear hyperbolic symmetric systems of equations driven by time-dependent vector fields which are only distributions in the time direction.