Feynman-Kac formula for iterated derivatives of the parabolic Anderson model

TL;DR

This paper derives a Feynman-Kac formula for the moments of iterated Malliavin derivatives of the parabolic Anderson model's solution, enabling new estimates for these moments using pinned Brownian motions.

Contribution

It introduces a novel Feynman-Kac representation for the moments of iterated derivatives in the parabolic Anderson model, linking stochastic analysis with Brownian motion techniques.

Findings

Established a Feynman-Kac formula for iterated derivatives

Provided moment estimates for derivatives of the solution

Linked Malliavin calculus with pinned Brownian motions

Abstract

The purpose of this paper is to establish a Feynman-Kac formula for the moments of the iterated Malliavin derivatives of the solution to the parabolic Anderson model in terms of pinned Brownian motions. As an application, we obtain estimates for the moments of the iterated derivatives of the solution.