Central limit theorem for finite and infinite dimensional diffusions in ergodic environments

In this article, we introduce a central limit theorem for the solution to a 1-dimensional stochastic heat equation with a random, ergodic non-linear term. Since it can be viewed as an infinite dimensional diffusion in random environment, we first summarize the ideas in the classical central limit theorem for finite dimensional diffusion in random environment. Afterward, we illustrate the strategy to extend these ideas to stochastic heat equations, and explain the main differences between finite and infinite dimensional models. Due to our result, a central limit theorem in L^1 sense with respect to the randomness of the environment holds and the limit distribution is a centered Gaussian law, whose covariance operator is explicitly described. Moreover, it concentrates only on the space of constant functions.