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

TL;DR

This paper investigates the spatial asymptotics of Feynman-Kac formulas for a parabolic Anderson model driven by nonhomogeneous, space-fractional rough Gaussian fields with measure-valued initial data, extending previous results.

Contribution

It provides new precise spatial asymptotics for Feynman-Kac formulas under broader conditions using Brownian bridge techniques.

Findings

Derived spatial asymptotics for the model

Extended the conditions for asymptotic analysis

Applied Brownian bridge approach to rough Gaussian fields

Abstract

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