# Normal approximation of the solution to the stochastic heat equation with L\'evy noise

## TL;DR

This paper establishes conditions under which solutions to the stochastic heat equation with Lévy noise converge in law to Gaussian solutions, extending finite-dimensional results to infinite-dimensional settings without assuming infinite divisibility.

## Contribution

It provides necessary and sufficient conditions for convergence of Lévy-driven solutions to Gaussian solutions in infinite dimensions, using martingale problem characterizations.

## Key findings

- Derived convergence conditions based on noise variances
- Extended finite-dimensional Lévy process results to infinite-dimensional case
- Characterized solutions via martingale problems in Sobolev spaces

## Abstract

Given a sequence $\dot{L}^{\varepsilon}$ of L\'evy noises, we derive necessary and sufficient conditions in terms of their variances $\sigma^2(\varepsilon)$ such that the solution to the stochastic heat equation with noise $\sigma(\varepsilon)^{-1} \dot{L}^\varepsilon$ converges in law to the solution to the same equation with Gaussian noise. Our results apply to both equations with additive and multiplicative noise and hence lift the findings of S. Asmussen and J. Rosi\'nski [J. Appl. Probab. 38 (2001) 482-493] and S. Cohen and J. Rosi\'nski [Bernoulli 13 (2007) 195-210] for finite-dimensional L\'evy processes to the infinite-dimensional setting without making distributional assumptions on the solutions such as infinite divisibility. One important ingredient of our proof is to characterize the solution to the limit equation by a sequence of martingale problems. To this end, it is crucial to view the solution processes both as random fields and as c\`adl\`ag processes with values in a Sobolev space of negative real order.