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

Abstract

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