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.

We analyze the one-dimensional periodic Kardar-Parisi-Zhang equation in the language of paracontrolled distributions, giving an alternative viewpoint on the seminal results of Hairer.Apart from a basic existence and uniqueness result for paracontrolled solutions to the KPZ equation we perform a thorough study of some related problems. We rigorously prove the links between KPZ equation, stochastic Burgers equation, and (linear) stochastic heat equation and also the existence of solutions starting from quite irregular initial conditions. We also build a partial link between energy solutions as introduced by Gonçalves and Jara and paracontrolled solutions.Interpreting the KPZ equation as the value function of an optimal control problem, we give a pathwise proof for the global existence of solutions and thus for the strict positivity of solutions to the stochastic heat equation.Moreover, we study Sasamoto-Spohn type discretizations of the stochastic Burgers equation and show that their limit solves the continuous Burgers equation possibly with an additional linear transport term. As an application, we give a proof of the invariance of the white noise for the stochastic Burgers equation which does not rely on the Cole-Hopf transform.

The Kardar-Parisi-Zhang (KPZ) equation is conjectured to universally describe the fluctuations of weakly asymmetric interface growth. Here we provide the first intrinsic well-posedness result for the KPZ equation on the real line by showing that its energy solutions as introduced by Gonçalves and Jara in [GJ14] and refined in [GJ13] are unique. Together with the convergence results of [GJ14] and many follow-up papers this establishes the weak KPZ universality conjecture for a wide class of models. Our proof builds on an observation of Funaki and Quastel [FQ15]. A remarkable consequence is that the energy solution to the KPZ equation is not equal to the Cole-Hopf solution, but it involves an additional drift t/12.