Temporal Hölder continuity of the parabolic Anderson model driven by a class of time-independent Gaussian fields with rough initial conditions

In this paper, we considered the parabolic Anderson model with a class of time-independent generalized Gaussian fields on ${\mathbb{R}}^d$, which included fractional white noise, Bessel field, massive free field, and other nonstationary Gaussian fields. Under the rough initial conditions, we constructed the Feynman-Kac formula as a solution in the Stratonovich integral by Brownian bridge, and then proved the Hölder continuity of the solution with respect to the time variable. As a comparison, we also studied the Hölder continuity under the regular initial conditions that $u_0\equiv C$ and $u_0\in C^{\kappa }({\mathbb{R}}^d)$ with $\kappa \in (0,1]$.