Convergence of space-discretised gKPZ via Regularity Structures

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.