Martingale-driven integrals and singular SPDEs

We consider multiple stochastic integrals with respect to càdlàg martingales, which approximate a cylindrical Wiener process. We define a chaos expansion, analogous to the case of multiple Wiener stochastic integrals, for these integrals and use it to show moment bounds. Key tools include an iteration of the Burkholder–Davis–Gundy inequality and a multi-scale decomposition similar to the one developed in Hairer and Quastel (Forum Math Pi 6:e3, 2018). Our method can be combined with the recently developed discretisation framework for regularity structures (Hairer and Matetski in Ann Probab 46(3):1651–1709, 2018, Erhard and Hairer in Ann Inst Henri Poincaré Probab Stat 55(4):2209–2248, 2019) to prove convergence of interacting particle systems to singular stochastic PDEs. A companion article (Grazieschiet al. in The dynamical Ising–Kac model in 3D converges to \(\Phi ^4_3\) ${\mathrm{\Phi }}_3^4$ , 2023. arXiv:2303.10242) applies the results of this paper to prove convergence of a rescaled Glauber dynamics for the three-dimensional Ising–Kac model near criticality to the \(\Phi ^4_3\) ${\mathrm{\Phi }}_3^4$ dynamics on a torus.