Small Ball Probabilities for the Stochastic Heat Equation on Compact Manifolds
Jiaming Chen
TL;DR
The paper develops small-ball probability estimates for the stochastic heat equation on arbitrary compact Riemannian manifolds, extending prior Euclidean and torus results to curved geometries. It uses an intrinsic Itô–Walsh framework with spatially colored noise built from the Laplace–Beltrami spectrum and spectral heat kernels, circumventing white-noise singularities in higher dimensions. A key proposition partitions space-time into blocks and derives upper and lower bounds for the relevant small-ball events, combining Gaussian approximations, stopping-time arguments, and the Gaussian correlation inequality. The main result delivers explicit exponential-type bounds in the small-noise limit, with regime-dependent exponents and a logarithmic correction at the critical line $\alpha=d/2$, highlighting how geometry and noise regularity shape extreme small-ball behavior. This work lays groundwork for a geometric theory of small-ball phenomena for SPDEs on manifolds and paves the way for further refinement across different curvature regimes and noise structures.
Abstract
We consider the stochastic heat equation on a compact smooth Riemannian manifold without boundary satisfying \begin{equation*} \partial_tu(t,x)=\frac{1}{2}Δ_Mu(t,x)+σ(t,x,u)\dot{W}(t,x),\quad (t,x)\in\mathbb{R}_+\times M, \end{equation*} where $\dot{W}$ is a centered Gaussian noise that is white in time and colored in space. Assuming that $σ$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,x)\equiv 0$.