Global Geometry within an SPDE Well-Posedness Problem

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

💡 Research Summary

This paper investigates the well-posedness and moment behavior of the Parabolic Anderson Model (PAM) on a closed Riemannian manifold, focusing on the profound influence of global geometry. The PAM is a stochastic heat equation driven by multiplicative noise, significant in probability theory and mathematical physics (e.g., directed polymers, KPZ equation).

The core challenge in extending classical well-posedness results to manifolds with measure-valued initial data is the singular nature of space-time white noise in dimensions d≥2. To overcome this, the authors construct an intrinsic family of Gaussian noises on the manifold, denoted W_α,ρ, which are white in time but colored in space. This noise is defined spectrally using the eigenfunctions of the Laplace-Beltrami operator, with its spatial covariance kernel G_α,ρ acting as an analogue of the Riesz kernel on Euclidean space. The well-posedness in the Itô-Walsh sense requires a regularity condition on the noise analogous to Dalang’s condition: α > (d-2)/2.

The main technical innovation lies in handling the non-uniqueness of minimizing geodesics on the manifold. The well-posedness proof relies on an iterative procedure whose key estimate involves an integral quantifying the concentration of measure for a Brownian bridge. This concentration is governed by a function F(s,t;x,y)(z) combining three distance terms. The minimizers of F lie on geodesics connecting the start and end points of the bridge. When two points are in each other’s cut locus (e.g., antipodal points on a sphere), multiple minimizing geodesics exist, making the analysis of F intractable with local arguments alone.

The authors resolve this global geometric difficulty by assuming the manifold has non-positive sectional curvature. This condition ensures that between any two points, there are only finitely many distance-minimizing geodesics. It allows the use of triangle comparison theorems to control the function F effectively, drawing parallels to the Euclidean case. This represents a novel instance where global differential geometry plays a decisive role in an SPDE well-posedness proof.

Under the non-positive curvature and Dalang conditions, the paper’s main theorem (Theorem 1.1) establishes:

1. For any finite initial measure μ, the PAM has a unique random field solution u(t,x). Furthermore, its p-th moment admits an exponentially growing upper bound in time: E