Martingale-driven approximations of singular stochastic PDEs

TL;DR

This paper develops a framework for approximating singular stochastic PDEs using martingale-driven stochastic integrals, enabling convergence proofs for discretizations of complex equations like KPZ and stochastic quantization.

Contribution

It introduces a novel approach to define and analyze stochastic integrals with respect to càdlàg martingales, facilitating convergence analysis of particle system discretizations for singular SPDEs.

Findings

Proved moment bounds and chaos expansions for martingale-driven integrals.

Established convergence of discretizations for 3D stochastic quantization and KPZ equations.

Abstract

We define multiple stochastic integrals with respect to c\`{a}dl\`{a}g martingales and prove moment bounds and chaos expansions, which allow to work with them in a way similar to Wiener stochastic integrals. In combination with the discretization framework of Erhard and Hairer (2017), our results give a tool for proving convergence of interacting particle systems to stochastic PDEs using regularity structures. As examples, we prove convergence of martingale-driven discretizations of the $3$-dimensional stochastic quantization equation and the KPZ equation.