Space-time discrete KPZ equation

TL;DR

This paper proves that a broad class of space-time discretizations of the KPZ equation converge to its continuous solution, extending the theory of regularity structures to discrete models with non-standard products.

Contribution

It develops a modified framework of regularity structures to handle space-time discretizations of the KPZ equation with non-standard products, proving convergence.

Findings

Discretizations converge to the continuous KPZ solution

Extended regularity structures to discrete models

Handled non-standard discretization of products

Abstract

We study a general family of space-time discretizations of the KPZ equation and show that they converge to its solution. The approach we follow makes use of basic elements of the theory of regularity structures [M. Hairer, A theory of regularity structures, Invent. Math. 2014] as well as its discrete counterpart [M. Hairer, K. Matetski, Discretizations of rough stochastic PDEs, 2015]. Since the discretization is in both space and time and we allow non-standard discretization for the product, the methods mentioned above have to be suitably modified in order to accommodate the structure of the models under study.