Martingale-driven integrals and singular SPDEs

Abstract

We consider multiple stochastic integrals with respect to c\`adl\`ag 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 arXiv:1512.07845. Our method can be combined with the recently developed discretisation framework for regularity structures arXiv:1511.06937, arXiv:1705.02836 to prove convergence of interacting particle systems to singular stochastic PDEs. A companion article titled "The dynamical Ising-Kac model in 3D converges to $\Phi^4_3$" 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 $\Phi^4_3$ dynamics on a torus.