# An invariance principle for the stochastic heat equation

## TL;DR

This paper presents an approximation approach for the stochastic heat equation driven by white noise, replacing the fractional Laplacian with a discrete generator and white noise with i.i.d. variables, providing a new proof of convergence.

## Contribution

It offers an alternative proof of the weak convergence of the scaled partition function of directed polymers to the stochastic heat equation, with convergence of all moments.

## Key findings

- Proves convergence of the scaled partition function to the stochastic heat equation.
- Provides an approximation scheme replacing fractional Laplacian with a discrete generator.
- Establishes convergence of all moments of the partition function.

## Abstract

We approximate the white-noise driven stochastic heat equation by replacing the fractional Laplacian by the generator of a discrete time random walk on the one dimensional lattice, and approximating white noise by a collection of i.i.d. mean zero random variables. As a consequence, we give an alternative proof of the weak convergence of the scaled partition function of directed polymers in the intermediate disorder regime, to the stochastic heat equation; an advantage of the proof is that it gives the convergence of all moments.