Optimal Extension to Sobolev Rough Paths

Abstract

We show that every mathbb {R}^{d}-valued Sobolev path with regularity α and integrability p can be lifted to a weakly geometric rough path in the sense of T. Lyons with exactly the same regularity and integrability, provided α > 1/p > 0. Moreover, we prove the existence of unique rough path lifts which are optimal w.r.t. strictly convex functionals among all possible rough path lifts given a Sobolev path. This paves a way towards classifying rough path lifts as solutions of optimization problems. As examples, we consider the rough path lift with minimal Sobolev norm and characterize the Stratonovich rough path lift of a Brownian motion as optimal lift w.r.t. a suitable convex functional. Generalizations of the results to Besov spaces are briefly discussed.