A Sobolev rough path extension theorem via regularity structures

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 provided $\alpha <1/p<1/3$. The novelty of our approach is its use of ideas underlying Hairer's reconstruction theorem generalized to a framework allowing for Sobolev models and Sobolev modelled distributions. Moreover, we show that the corresponding lifting map is locally Lipschitz continuous with respect to the inhomogeneous Sobolev metric.

arXiv commentThis manuscript is mainly part of the preprint arXiv:1811.05173v2, which is now divided in two separate works. Typos are corrected, to appear in ESAIM:Probability and Statistics