Small Ball Probabilities for the Fractional Stochastic Heat Equation Driven by a Colored Noise

Abstract

We consider the fractional stochastic heat equation on the $d$-dimensional torus ${\mathbb{T}}^d:={\left[-\frac{1}{2},\frac{1}{2}\right]}^d$, $d\ge 1$, 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 $\hat{I}\pm \in (1,2]$ and $\dot{F}(t,\text{x})$ is a generalized Gaussian noise which is white in time and colored in space. Assuming that $\ddot{I}ƒ$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,\text{x})\equiv 0$.

Published in EJP