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 B(H )for H > (1) (4), we prove that the drift may be taken to be kappa > 0 H & ouml;lder continuous and bounded for kappa > (3)(2) - (1) (2H) . A flow transform of the equation and Malliavin 1 calculus for Gaussian rough paths are used to achieve such a result.