Spatial Hölder continuity for the parabolic Anderson model with the singular initial conditions

Under the singular initial conditions, we consider the parabolic Anderson model driven by the time-independent generalized Gaussian fields, which include some frequently-used non-stationary Gaussian fields. When the initial value u0 belongs to the weighted Besov space with negative regularity Bq,∞−κ,wλ, it is proved that there exists a unique pathwise solution for the model in the Young sense. Moreover, if u0 also satisfies the measure-valued initial condition, by the Feynman-Kac formula based on Brownian bridge, we find that the solution owns a spatially Hölder continuous modification.