Solving the KPZ Equation

Abstract

We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a "universal" measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction completely bypasses the Cole-Hopf transform, thus laying the groundwork for proving that the KPZ equation describes the fluctuations of systems in the KPZ universality class.As a corollary of our construction, we obtain very detailed new regularity results about the solution, as well as its derivative with respect to the initial condition. Other byproducts of the proof include an explicit approximation to the stationary solution of the KPZ equation, a well-posedness result for the FokkerPlanck equation associated to a particle diffusing in a rough space-time dependent potential, and a new periodic homogenisation result for the heat equation with a space-time periodic potential. One ingredient in our construction is an example of a non-Gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge.

Stochastic Partial Differential Equations: An Introduction

“…a power of the solution, have to be defined by renormalization techniques (see [2] and also [1]). Recently, a breakthrough has been achieved in this direction by Martin Hairer in [44] (for which, together with his work on the KPZ-equation [43] and other beautiful results in the field, he was awarded the Fields Medal in 2014). In [44] he develops a whole theory to define and (locally) solve nonlinear SPDEs in a reformulated framework.…”

Introduction to regularity structures

“…meaning to ill-posed stochastic PDEs that arise naturally when trying to describe the macroscopic behaviour of models from statistical mechanics near criticality. One example of such an equation is the KPZ equation arising as a natural model for one-dimensional interface motion (Kardar et al, 1986;Bertini and Giacomin, 1997;Hairer, 2013): distributions. One caveat that also has to be dealt with in the latter case is that the limiting process may in principle explode in finite time for some instances of the driving noise.…”

The Parabolic Anderson Model

“…This topic recently received a particular push when the theory of regularity structures was developed so far that a solution to the KPZ equation could be constructed [Hai13]; see also [HaiLab15b]. Actually, the KPZ equation and related models are currently one of the hottest research subjects in probability and related fields, but clearly outside the scope of this text.…”

“…Only since very recently, techniques for overcoming these problems are being developed, and there is currently a high activity in that direction in the spirit of the theory of rough paths and novel methods like the theory of regularity structures [Hai13] and paracontrolled distributions [GubImkPer12]. Constructing solutions to the PAM is one of the few major application fields and test cases for these methods, the other prominent examples being the KPZ equation and the 4 -model.…”