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

We consider the fractional stochastic heat equation on the $d$-dimensional torus ${\mathbb{T}}^d:={[-\frac{1}{2},\frac{1}{2}]}^d$, $d\ge 1$, with periodic boundary conditions: $${\mathrm{\partial }}_{t}u(t,\mathbf{\text{x}})=-(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2}u(t,\mathbf{\text{x}})+\sigma (t,\mathbf{\text{x}},u)\dot{F}(t,\mathbf{\text{x}})\mathbf{\text{x}}\in {\mathbb{T}}^d,t\in {\mathbb{R}}_{+},$$ where $\alpha \in (1,2]$ and $\dot{F}(t,\mathbf{\text{x}})$ is a generalized Gaussian noise which is white in time and colored in space. Assuming that $\sigma$ is Lipschitz in $u$ and uniformly bounded, we estimate small ball probabilities for the solution $u$ when $u(0,\mathbf{\text{x}})\equiv 0$.