R. Catellier and M. Gubinelli
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 $\rho$-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–Hölder 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.
R. Catellier and M. Gubinelli. Averaging along irregular curves and regularisation of ODEs. Papers of Massimiliano Gubinelli, 2012, version 4. https://mgubi.org/papers/paper/averaging-irregular-curves/
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