Spatial asymptotics for the Feynman–Kac formulas driven by time-dependent and space-fractional rough Gaussian fields with the measure-valued initial data

We consider the continuous parabolic Anderson model with Gaussian fields under measure-valued initial conditions, covariances of which are homogeneous or nonhomogeneous in time and fractional rough space. mainly study spatial behaviors for Feynman–Kac formulas Stratonovich sense. Benefited from application formula based on Brownian bridge, precise asymptotics can be obtained more general conditions than it previous literature.