Stochastic parabolic Anderson model with time-homogeneous generalized potential: Mild formulation of solution

TL;DR

This paper develops a mild formulation for the stochastic parabolic Anderson model with Gaussian potential, enabling analysis of solution regularity and exploring interpretations with pathwise and Wick products.

Contribution

It introduces a mild solution framework for the model and highlights the importance of a modified Schauder estimate for existence and uniqueness.

Findings

Mild formulation facilitates regularity analysis of solutions.

Modified Schauder estimates are key for proving existence and uniqueness.

Comparison between pathwise and Wick product interpretations is clarified.

Abstract

A mild formulation for stochastic parabolic Anderson model with time-homogeneous Gaussian potential suggests a way of defining a solution to obtain its optimal regularity. Two different interpretations in the equation or in the mild formulation are possible with usual pathwise product and the Wick product: the usual pathwise interpretation is mainly discussed. We emphasize that a modified version of parabolic Schauder estimates is a key idea for the existence and uniqueness of a mild solution. In particular, the mild formulation is crucial to investigate a relation between the equation with usual pathwise product and the Wick product.