Small Ball Probabilities for the Fractional Stochastic Heat Equation Driven by a Colored Noise
Jiaming Chen
TL;DR
This work analyzes small ball probabilities for the mild solution u of the fractional stochastic heat equation on the torus with space-colored noise. By partitioning space into ε-boxes and time into short intervals, the authors construct a Gaussian-approximation framework and use measure-change techniques to derive sharp exponential bounds for P(sup_{0≤t≤T, x∈ o ext{T}^d} |u(t,x)| ≤ ε). The main contributions are (i) a precise two-sided bound in the 1D regime with 2β≤α, and (ii) a general upper-bound (and conditional lower-bound) formula in higher dimensions or when 2β>α, all depending on α, β, d, and the noise structure Λ. The methods combine heat-kernel estimates, stochastic convolution regularity, and Gaussian comparison inequalities to extend small-ball analyses to multiplicative, non-Gaussian SPDEs with colored spatial noise.
Abstract
We consider the fractional stochastic heat equation on the $d$-dimensional torus $\mathbb{T}^d:=\left[-\frac{1}{2},\frac{1}{2}\right]^d$, $d\geq 1$, with periodic boundary conditions: \[ \partial_t u(t,\textbf{x})= -(-Δ)^{α/2}u(t,\textbf{x})+σ(t,\textbf{x},u)\dot{F}(t,\textbf{x})\quad \textbf{x}\in \mathbb{T}^d,t\in\mathbb{R}_+ ,\] where $α\in(1,2]$ and $\dot{F}(t,\textbf{x})$ is a generalized Gaussian noise which 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 $u(0,\textbf{x})\equiv 0$.