Well-posedness of Rough Differential Equations at the Krylov-Röckner Integrability Threshold

We study rough differential equations (RDEs) driven by Brownian (Itô) rough paths with uniformly locally integrable drifts. The diffusion coefficient is assumed to be uniformly elliptic, of class $C^{2+}$ in space and of bounded variation in time, while the drift is $q$-integrable in time and $p$-integrable in space, uniformly on balls of a fixed radius, for some $p,q$ satisfying the Krylov--Röckner condition $\frac dp+\frac2q<1$. We show that, for almost every realization of the Itô rough path lift of the Brownian motion, such an RDE has a unique solution for every initial value. In other words, for almost every realization of the Itô lift, the corresponding rough differential equation has exactly one solution in the class of Davie-type solutions, for every initial value; this is the appropriate notion of path-by-path uniqueness for the corresponding stochastic differential equation (SDE) with multiplicative noise. The proof follows the strategy of Shaposhnikov (2016). The principal new ingredients are the stability of RDEs with bounded variation perturbation and Davie-type estimates for the flow of the driftless equation with non-constant diffusion coefficients, which we derive from known maximal regularity estimates for the corresponding parabolic partial differential equations (PDEs).