Martingale-driven integrals and singular SPDEs

• 成果类型: Article
• 署名作者: Grazieschi, P.; Matetski, K.; Weber, H.
• 署名单位: University of Bath; Michigan State University; University of Munster
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-024-01311-2
• 发表日期: 2024
• 页码: 1063-1120
• 关键词: stochastic burgers; fluctuations; equation; MODEL

摘要

We consider multiple stochastic integrals with respect to c & agrave;dl & agrave;g 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 Poincar & eacute; 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 34\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi <^>4_3$$\end{document}, 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 34\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi <^>4_3$$\end{document} dynamics on a torus.