Averaging along irregular curves and regularisation of ODEs

We consider the ordinary differential equation (ODE) $dx_{t} =b(t,x_{t} ) dt+ dw_{t}$ where $w$ is a continuous driving function and $b$ is a time-dependent vector field which possibly is only a distribution in the space variable. We quantify the regularising properties of an arbitrary continuous path $w$ on the existence and uniqueness of solutions to this equation. In this context we introduce the notion of $\rho$-\tmtextit{irregularity} and show that it plays a key role in some instances of the regularisation by noise phenomenon. In the particular case of a function $w$ sampled according to the law of the fractional Brownian motion of Hurst index $H \in (0,1)$, we prove that almost surely the ODE admits a solution for all $b$ in the Besov-H\~A{\P}lder space $B^{\alpha+1}_{\infty , \infty}$ with $\alpha >-1/2H$. If $\alpha >1-1/2H$ then the solution is unique among a natural set of continuous solutions. If $H>1/3$ and $\alpha >3/2-1/2H$ or if $\alpha >2-1/2H$ then the equation admits a unique Lipschitz flow. Note that when $\alpha <0$ the vector field $b$ is only a distribution, nonetheless there exists a natural notion of solution for which the above results apply.