Regularization by noise for rough differential equations driven by 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 $H>\frac{1}{4}$, we prove that the drift may be taken to be $\kappa >0$ Hölder continuous and bounded for $\kappa >\frac{3}{2}-\frac{1}{2H}$. A flow transform of the equation and Malliavin calculus for Gaussian rough paths are used to achieve such a result.

Subtential changes, especially in Section 4

Cited by in corpus (1)

• Path-by-path uniqueness for stochastic differential equations under Krylov-Röckner condition