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

We consider the rough differential equation with drift driven by a Gaussian geometric rough path. Under natural conditions on the rough path, namely nondeterminism, 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 BH for H>14, we prove that the drift may be taken to be κ>0 Hölder continuous and bounded for κ>32−12H. A flow transform of the equation and Malliavin calculus for Gaussian rough paths are used to achieve such a result.