PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES

Massimiliano Gubinelli,Peter Imkeller,Nicolas Perkowski

doi:10.1017/fmp.2015.2

Abstract

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.

Highlights

We introduce the notion of paracontrolled distribution and show how to use it to give a meaning to and solve partial differential equations (PDEs) involving nonlinear operations on generalized functions
We combine the idea of controlled paths, introduced in [Gub04], with the paraproduct and the related paradifferential calculus introduced by Bony [Bon81], in order to develop a nonlinear theory for a certain class of distributions
The approach presented here works for generalized functions defined on an index set of arbitrary dimension and constitutes a flexible and lightweight generalization of Lyons’ rough path theory [Lyo98]

Summary

Introduction

We introduce the notion of paracontrolled distribution and show how to use it to give a meaning to and solve partial differential equations (PDEs) involving nonlinear operations on generalized functions. Hairer’s notion of smoothness induces a natural topology in which the solutions to semilinear SPDEs depend continuously on the driving signal This approach is very general, and it allows us to handle more complicated problems than the ones we are currently able to treat in the paracontrolled approach. Together with the equation u = F(u) ≺ θ + u , this completely describes the solution and allows us to obtain an a priori estimate on u in terms of (u0, ξ α−1, θ ◦ ξ 2α−1) With this estimate at hand, it is relatively straightforward to show that, if F ∈ Cb3, u depends continuously on the data (u0, ξ, θ ◦ ξ ), so that we can pass to the limit in (5) and make sense of the solution to (4) for irregular ξ ∈ C α−1 as long as α > 1/3.

Rough differential equations

Rough Burgers equation

A generalized parabolic Anderson model

The equivalence of moments for

Relation with regularity structures