We study existence and uniqueness of solutions to the equation dXt = b(Xt)dt + dBt, where b is a distribution in some Besov space and B is a fractional Brownian motion with Hurst parameter H 1/2. First, the equation is understood as a nonlinear Young integral equation. The integral is constructed in a p-variation space, which is well suited when b is a nonnegative (or nonpositive) distribution. Based on the Besov regularity of b, a condition on H is given so that solutions to the equation exist. The construction is deterministic, and B can be replaced by a deterministic path w which has a sufficiently smooth local time.We then consider probabilistic notions of weak and strong solution to this SDE. In particular, when b is a nonnegative finite measure, we show the existence of weak solutions for H < √ 2 − 1. We prove that solutions can also be understood as limits of strong solutions when b is regularised, which is used to establish pathwise uniqueness and existence of a strong solution when H 1/4. The proofs involve fine properties of the local time of the fractional Brownian motion, as well as new regularising properties of this process which are established using the stochastic sewing Lemma.

We consider the Stochastic Differential EquationWe give an example of a drift b such that there does not exist a weak solution, but there exists a solution for almost every realization of the Brownian motion B. We also give an explicit example of a drift such that the SDE has a pathwise unique weak solution, but pathby-path uniqueness (i.e. uniqueness of solutions to the ODE for almost every realization of the Brownian motion) is lost. These counterexamples extend the results obtained in [22] to dimension d = 1.