Title: Spatial ergodicity and quantitative central limit theorems for the stochastic heat equation.

Abstract:  Consider the d-dimensional stochastic heat equation driven by a Gaussian noise, which is white in time and it has an homogeneous spatial covariance. In  this talk we will present some recent results on the ergodicity of the solution in the space variable and total variation estimates for renormalized spatial averages. These results are obtained using techniques of Malliavin calculus. We will review the main ingredients of the proofs, which are Poincaré-type inequalities and the Stein-Malliavin approach to estimate the total variation distance to a standard normal distribution.