Gaussian Rough Paths Lifts via Complementary Young Regularity

TL;DR

This paper demonstrates that controlled complementary Young regularity suffices to lift Gaussian processes to rough paths without assuming 2D variation regularity of the covariance, simplifying proofs and improving convergence results.

Contribution

It introduces a novel approach using the Poincaré inequality on Wiener space to achieve Gaussian rough path lifts under minimal assumptions.

Findings

Achieves Gaussian rough path lifts without 2D covariance regularity assumptions.

Provides cleaner proofs with optimal convergence rates.

Shows convergence of random Fourier series in rough paths metrics.

Abstract

Inspired by recent advances in singular SPDE theory, we use the Poincar\'e inequality on Wiener space to show that controlled complementary Young regularity is sufficient to obtain Gaussian rough paths lifts. This allows us to completely bypass assumptions on the 2D variation regularity of the covariance and, as a consequence, we obtain cleaner proofs of approximation statements (with optimal convergence rates) and show the convergence of random Fourier series in rough paths metrics under minimal assumptions on the coefficients (which are sharper than those in the existent literature).