Convergence of space-discretised gKPZ via Regularity Structures

TL;DR

This paper proves the convergence of a space-discretised generalized KPZ equation using regularity structures, extending previous results and marking a significant step towards a comprehensive convergence theory for discrete models.

Contribution

It introduces the first convergence result for a discrete generalized KPZ equation using regularity structures, expanding the applicability of the theory to a broad class of models.

Findings

Established convergence of discrete gKPZ without fixing spatial dimension

Extended regularity structures to handle discrete models of gKPZ

Paved the way for a general convergence framework for discrete stochastic PDEs

Abstract

In this work, we show a convergence result for the discrete formulation of the generalised KPZ equation $\partial_t u = (\Delta u) + g(u)(\nabla u)^2 + k(\nabla u) + h(u) + f(u)\xi_t(x)$, where the $\xi$ is a real-valued random field, $\Delta$ is the discrete Laplacian, and $\nabla$ is a discrete gradient, without fixing the spatial dimension. Our convergence result is established within the discrete regularity structures introduced by Hairer and Erhard [arXiv:1705.02836]. We extend with new ideas the convergence result found in [arXiv:2103.13479] that deals with a discrete form of the Parabolic Anderson model driven by a (rescaled) symmetric simple exclusion process. This is the first time that a discrete generalised KPZ equation is treated and it is a major step toward a general convergence result that will cover a large family of discrete models.