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

Abstract

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] $.

Keywords

• parabolic anderson model
• feynman-kac formula
• generalized gaussian field
• brownian bridge
• hölder continuity
• measure-valued initial data