Global Geometry within an SPDE Well-Posedness Problem

TL;DR

This paper investigates the well-posedness of the parabolic Anderson model on curved spaces, revealing how global geometry and curvature influence solution existence and growth, with new insights into Gaussian noise and geodesic non-uniqueness.

Contribution

It introduces a framework for analyzing SPDEs on curved manifolds, highlighting the role of non-positive curvature and global geometry in establishing well-posedness.

Findings

Well-posedness depends on non-positive curvature and noise regularity.

Global geometry affects the uniqueness of solutions.

Solutions exhibit exponential growth in moments over time.

Abstract

On a closed Riemannian manifold, we construct a family of intrinsic Gaussian noises indexed by a regularity parameter $\alpha\geq0$ 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 $\alpha$ 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.