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 ${\partial }_{\mathit{t}}\mathit{u}=(\mathrm{\Delta }\mathit{u})\mathbf{+}\mathit{g}(\mathit{u}){(\nabla \mathit{u})}^2\mathbf{+}\mathit{k}(\nabla \mathit{u})\mathbf{+}\mathit{h}(\mathit{u})\mathbf{+}\mathit{f}(\mathit{u}){\mathit{\xi }}_{\mathit{t}}(\mathit{x})$, where ξ is real-valued, Δ is the discrete Laplacian, and ∇ is a discrete gradient, without fixing the spatial dimension. Our convergence result is established within the discrete regularity structures introduced by Hairer and Erhard (Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019) 2209–2248). We extend with new ideas the convergence result found in (Comm. Pure Appl. Math. 77 (2024) 1065–1125) 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.