[论文解读] Small Ball Probabilities for the Stochastic Heat Equation on Compact Manifolds

Jiaming Chen|arXiv (Cornell University)|Jan 28, 2026

Stochastic processes and financial applications被引用 0

一句话总结

论文在紧致Riemannian流形上，对Ito-Walsh解的随机热方程在时间上白噪、空间上有色噪声的条件下，给出小球概率估计，前提是扩散系数满足 Lipschitz 性与有界性。

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}_+ imes 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 $u(0,x)\equiv 0$.

研究动机与目标

• Motivate and study SPDEs of parabolic type on compact Riemannian manifolds without boundary.
• Develop an intrinsic framework for spatially colored noise that replaces Euclidean Fourier analysis with Laplace–Beltrami spectral decomposition.
• Obtain small ball probability estimates for non-Gaussian solutions of the stochastic heat equation under Dalang’s condition.

提出的方法

• Formulate the stochastic heat equation on a compact manifold using the Laplace–Beltrami operator and heat kernel.
• Introduce a spectral-based Gaussian noise with covariance defined via Laplace–Beltrami eigenfunctions and a parameter alpha, creating the Hilbert space H^{alpha, rho}.
• Impose Lipchitz and uniform boundedness assumptions on the diffusion coefficient sigma to ensure well-posedness.
• Decompose the domain into nested geodesic balls and time intervals to construct events controlling the solution, and use Gaussian approximation arguments for upper bounds.
• Apply Gaussian correlation inequality and change-of-measure arguments to obtain lower bounds for small ball probabilities.
• Leverage heat kernel estimates and noise term regularity results to derive explicit dependence of bounds on alpha, d, and the small parameter epsilon.

实验结果

研究问题

• RQ1 What are the small ball probabilities for the solution to the stochastic heat equation on a compact manifold under spatially colored noise?
• RQ2 How does the geometry of M and the noise parameter alpha influence the decay rate in small ball probabilities?
• RQ3 Can one obtain matching upper and lower bounds for P(sup_{t≤T, x∈M}|u(t,x)|<ε) in this geometric setting?
• RQ4 How does one adapt Euclidean Fourier-based methods to manifold settings via spectral analysis of the Laplace–Beltrami operator?
• RQ5 What is the role of Dalang’s condition in ensuring existence and regularity needed for small ball analysis on manifolds?

主要发现

• Existence of small ball probability estimates for the non-Gaussian solution under Lipschitz and nondegenerate diffusion coefficient assumptions.
• Upper bounds exhibit exponential decay of the form exp(-C T / ε^{(2d+4)/h}) with h = min(1, 2α − d + 2), reflecting the interplay between dimension, noise regularity, and manifold geometry.
• Lower bounds show exponential decay of the form exp(-C T / ε^{(2d+4)/h}) up to constants, achieving a nontrivial rate matching the upper bounds up to regime-dependent refinements.
• The analysis reveals a distinct exponent behavior across regimes defined by α relative to d/2, including a logarithmic correction at the critical α = d/2.
• The framework replaces Euclidean Fourier analysis with spectral analysis of the Laplace–Beltrami operator, enabling intrinsic, coordinate-free small ball estimates on manifolds.
• The approach handles the boundary-difficult regime α = d/2, which is inaccessible in prior flat-geometry studies.