Differential Equations driven by \Pi-rough paths

TL;DR

This paper extends the theory of rough paths by introducing geometric -rough paths with inhomogeneous smoothness, proving integral existence under weaker conditions, and establishing existence and uniqueness of solutions for associated differential equations.

Contribution

It introduces geometric -rough paths, broadening rough path theory, and provides new conditions for integral existence and solution uniqueness in differential equations driven by these paths.

Findings

Existence of integrals under weaker conditions.

Sufficient conditions for solution existence and uniqueness.

Extension of rough path theory to inhomogeneous smoothness.

Abstract

This paper revisits the concept of rough paths of inhomogeneous degree of smoothness (geometric \Pi-rough paths in our terminology) sketched by Lyons ("Differential equations driven by rough signals", Revista Mathematica Iber. Vol 14, Nr. 2,215-310, 1998). Although geometric \Pi-rough paths can be treated as p-rough paths for a sufficiently large p and the theory of integration of Lip-\gamma one-forms (\gamma>p-1) along geometric p-rough paths applies, we prove the existence of integrals of one-forms under weaker conditions. Moreover, we consider differential equations driven by geometric \Pi-rough paths and give sufficient conditions for existence and uniqueness of solution.