Ergodicity and Gaussian fluctuations of stochastic heat equations

In this talk, I will present some recent results about the ergodicity of the stochastic heat equation driven by centred Gaussian noise that is white in time and coloured in space. A sufficient condition for ergodicity and classify the invariant measures in terms of their expectations will be introduced. Under this regime, we also establish a Gaussian fluctuation result under diffusive scaling. Furthermore, we show that the corresponding scaling limit is governed by the Edwards–Wilkinson equation. This talk is based on joint work with Le Chen, Alex Dunlap, Cheng Ouyang, and Samy Tindel.