Paracontrolled distributions and singular PDEs

TL;DR

This paper introduces a novel approach combining paradifferential calculus and controlled rough paths to analyze singular partial differential equations, demonstrating its effectiveness on models driven by fractional Brownian motion and white noise.

Contribution

It presents a new framework for studying singular PDEs that integrates techniques from paradifferential calculus and rough path theory, applicable to complex stochastic models.

Findings

Successfully applied to fractional Brownian motion driven equations

Effectively analyzed SPDEs with space-time white noise

Provided new insights into non-linear parabolic Anderson models

Abstract

We introduce an approach to study certain singular 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 like differential equations driven by fractional Brownian motion, a fractional Burgers type SPDE driven by space-time white noise, and a non-linear version of the parabolic Anderson model with a white noise potential.