Optimal extension to Sobolev rough paths

TL;DR

This paper establishes conditions under which Sobolev paths can be uniquely lifted to Sobolev rough paths, introducing optimal lifts based on convex functionals, including the minimal Sobolev norm and the Stratonovich lift for Brownian motion.

Contribution

It proves the existence and uniqueness of optimal Sobolev rough path lifts, extending the theory to Besov spaces and characterizing the Stratonovich lift as optimal.

Findings

Sobolev paths can be lifted to Sobolev rough paths when lpha > 1/p.

Unique optimal lifts exist with respect to convex functionals.

The Stratonovich lift of Brownian motion is characterized as an optimal lift.

Abstract

We show that every $\mathbb{R}^d$-valued Sobolev path with regularity $\alpha$ and integrability $p$ can be lifted to a Sobolev rough path in the sense of T. Lyons provided $\alpha >1/p>0$. 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. 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. to a suitable convex functional. Generalizations of the results to Besov spaces are briefly discussed.