Global Existence of Geometric Rough Flows

TL;DR

This paper establishes conditions under which rough differential equations on smooth manifolds have solutions that exist for all time, extending fundamental results to a geometric setting.

Contribution

It provides new sufficient conditions involving bounded covariant derivatives for the global existence of solutions to rough differential equations on manifolds.

Findings

Sufficient conditions for global solutions on manifolds.

Extension of fundamental rough path results to geometric settings.

Conditions involve bounded covariant derivatives and a complete Riemannian metric.

Abstract

In this paper we consider rough differential equations on a smooth manifold $\left( M\right) .$ The main result of this paper gives sufficient conditions on the driving vector-fields so that the rough ODE's have global (in time) solutions. The sufficient conditions involve the existence of a complete Riemannian metric $\left( g\right) $ on $M$ such that the covariant derivatives of the driving fields and their commutators to a certain order (depending on the roughness of the driving path) are bounded. Many of the results of this paper are generalizations to manifolds of the fundamental results in \cite{Bailleul2015a}.