A geometric view on rough differential equations

Abstract

Our contribution to the theory of rough paths is twofold. On the one hand we introduce tree algebras over topological vector spaces which yields a framework for branched rough path in infinite dimensional spaces. In particular, we derive a change-of-variables formula in this settings from a purely combinatorial argument and we revisit the relationship between geometric, branched and general rough paths and we discuss to what extent each of them can be used to drive differential equations on manifolds.

On the other hand we discuss global solutions of rough differential equations on manifolds. We show that instead of considering bounds for the full higher order derivatives of the vector fields it is enough to look at Lie brackets. In our proofs we extend methods from Riemannian geometry that have been used to obtain characterisations of complete vector fields on manifolds. This leads to a unified result for rough differential equations that captures examples such as bounded vector fields, vector fields of linear growth or complete commuting vector fields all at once. In particular, our result holds for vector spaces and it improves existing non-explosion criteria in this case. We present an example of an equation which has global solutions for all initial values but where existing methods were not sufficient to give a proof.

Finally, we give an outlook as to where the new results may lead to further developments.