KPZ reloaded

TL;DR

This paper provides a new analytical framework for the one-dimensional KPZ equation using paracontrolled distributions, establishing existence, uniqueness, and connections to related stochastic equations, with applications to discretizations and invariance properties.

Contribution

It introduces an alternative paracontrolled approach to KPZ, linking it to stochastic Burgers and heat equations, and explores discretizations and invariance without Cole-Hopf transform.

Findings

Proved existence and uniqueness of paracontrolled solutions to KPZ.

Established links between KPZ, stochastic Burgers, and stochastic heat equations.

Demonstrated invariance of white noise for stochastic Burgers without Cole-Hopf transform.

Abstract

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 deriving 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 Goncalves 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.