Global geometry within an SPDE well-posedness problem

• 成果类型: Article; Early Access
• 署名作者: Chen, Hongyi; Ouyang, Cheng
• 署名单位: University of Illinois System; University of Illinois Chicago; University of Illinois Chicago Hospital
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-025-01460-y
• 发表日期: 2025-12-23
• 关键词: stochastic heat-equation; moments; burgers; driven; noise

摘要

On a closed Riemannian manifold, we construct a family of intrinsic Gaussian noises indexed by a regularity parameter alpha >= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \ge 0$$\end{document} to study the well-posedness of the parabolic Anderson model. We show that with rough initial conditions, the equation is well-posed assuming non-positive curvature with a condition on alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} similar to that of Riesz kernel-correlated noise in Euclidean space. Non-positive curvature was used to overcome a new difficulty introduced by non-uniqueness of geodesics in this setting, which required exploration of global geometry. The well-posedness argument also produces exponentially growing in time upper bounds for the moments. Using the Feynman-Kac formula for moments, we also obtain exponentially growing in time second moment lower bounds for our solutions with bounded initial condition.