2025/02/06 by Hongyi Chen, Cheng Ouyang, Chen, Hongyi +1 · 1 citation
Engineering · #Advanced Control Systems Optimization #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2502.04572
openalex publication_date 2025/02/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
On a closed Riemannian manifold, we construct a family of intrinsic Gaussian noises indexed by a regularity parameter α≥0 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 α 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 Feynman-Kac formula for moments, we also obtain exponentially growing in time second moment lower bounds for our solutions with bounded initial condition.