Convergence of space-discretised gKPZ via Regularity Structures

Yvain Bruned, Usama Nadeem

Published 2022-07-20Version 1

In this article, we propose a semi-general extension to the convergence result found in (Erhard & Hairer, 2021). In that paper, the authors showed that a discrete form of the Parabolic Anderson model driven by a (rescaled) symmetric simple exclusion process on the torus of dimension $3$, can be renormalised in such a manner that the sequence of solutions converges in law to a continuous process that solves a (renormalised) version of the PAM driven by a generalised Ornstein-Uhlenbeck process. We will showcase our extension by proving a similar 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 major contribution lies in extending the very general bounds on labelled rooted binary trees presented in (Hairer & Quastel, 2018), and further proposing a new renormalisation procedure that involves local transformations on the level of the trees that makes it possible to treat some divergences that were not tractable under the framework in (Erhard & Hairer, 2021). While this procedure falls short of complete generality as provided by (Bruned et al., 2019) for singular SPDEs in the continuum, we believe this is a possible intermediate step in that direction for the discrete case.