We formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints. This allows us to define a good notion of integral with respect to irregular paths with Ho¨lder exponent greater than 1=3 (e.g. samples of Brownian motion) and study the problem of the existence, uniqueness and continuity of solution of differential equations driven by such paths. We recover Young's theory of integration and the main results of Lyons' theory of rough paths in Ho¨lder topology. r

We introduce an approach to study certain singular partial differential equations (PDEs) which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems such as differential equations driven by fractional Brownian motion, a fractional Burgers-type stochastic PDE (SPDE) driven by space-time white noise, and a nonlinear version of the parabolic Anderson model with a white noise potential.