Two-point correlation function and Feynman-Kac formula for the stochastic heat equation

TL;DR

This paper derives an explicit two-point correlation function for the stochastic heat equation on the real line, simplifies moment bounds, and validates the Feynman-Kac formula for measure-valued initial data.

Contribution

It provides a new explicit formula for the two-point correlation function and confirms the Feynman-Kac representation for solutions with measure-valued initial conditions.

Findings

Explicit two-point correlation function derived

Simplified bounds for p-th moments

Validated Feynman-Kac formula for measure-valued initial data

Abstract

In this paper, we obtain an explicit formula for the two-point correlation function for the solutions to the stochastic heat equation on $\mathbb{R}$. The bounds for $p$-th moments proved in [3] are simplified. We validate the Feynman-Kac formula for the $p$-point correlation function of the solutions to this equation with measure-valued initial data.