Regularization by noise for rough differential equations driven by Gaussian rough paths

TL;DR

This paper proves the well-posedness of rough differential equations driven by Gaussian rough paths with irregular drifts, using flow transforms and Malliavin calculus, especially for fractional Brownian motion with Hurst parameter greater than 1/4.

Contribution

It establishes path-by-path well-posedness for Gaussian rough differential equations with poorly regular drifts, extending previous results to fractional Brownian motion with H>1/4.

Findings

Well-posedness under non-determinism and ellipticity conditions

Drift regularity can be as low as Hölder continuous with specific bounds

Application of Malliavin calculus to Gaussian rough paths

Abstract

We consider the rough differential equation with drift driven by a Gaussian geometric rough path. Under natural conditions on the rough path, namely non-determinism, and uniform ellipticity conditions on the diffusion coefficient, we prove path-by-path well-posedness of the equation for poorly regular drifts. In the case of the fractional Brownian motion $B^H$ for $H>\frac14$, we prove that the drift may be taken to be $\kappa>0$ H\"older continuous and bounded for $\kappa>\frac32 - \frac1{2H}$. A flow transform of the equation and Malliavin calculus for Gaussian rough paths are used to achieve such a result.