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Global geometry within an SPDE well-posedness problem
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DOI：10.1007/s00440-025-01460-y.png)
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Abstract
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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.
Keywords:
STOCHASTIC HEAT-EQUATION
MOMENTS
BURGERS
DRIVEN
NOISE
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