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Motivated by the recent advances in the theory of stochastic partialdifferential equations involving nonlinear functions of distributions, like theKardar-Parisi-Zhang (KPZ) equation, we reconsider the unique solvability ofone-dimensional stochastic differential equations, the drift of which is adistribution, by means of rough paths theory. Existence and uniqueness areestablished in the weak sense when the drift reads as the derivative of aH{\"o}lder continuous function. Regularity of the drift part is investigatedcarefully and a related stochastic calculus is also proposed, which makes thestructure of the solutions more explicit than within the earlier framework ofDirichlet processes.

Rough paths and 1d sde with a time dependent distributional drift. Application to polymers