Moments of the SHE under delta initial measure

TL;DR

This paper rigorously proves contour integral formulas for the moments of the stochastic heat equation with delta initial measure, confirming conjectures and establishing a connection with the Airy point process.

Contribution

It provides a rigorous proof of conjectured moment formulas for SHE with delta initial condition, utilizing a correspondence with the Airy point process.

Findings

Confirmed conjectured moment formulas for SHE

Established a link between SHE and Airy point process

Validated previous heuristic and numerical results

Abstract

We give a rigorous proof of the contour integral formulas of the moments of the stochastic heat equation (SHE) started from the delta initial measure at the origin. These formulas were conjectured in [BC14] (see also [CDR10, Dot10]). Our proof is based on a correspondence between the SHE and the Airy point process which was proved in [BG16, Theorem 1] using the formula of [ACQ11, Theorem 1.1].