A Liouville theorem for stationary and ergodic ensembles of parabolic systems

In this talk, which is based on joint work with Peter Bella and Alberto Chairini, I will discuss a first-order Liouville theorem for random ensembles of uniformly parabolic systems under the qualitative assumptions of stationarity and ergodicity. The method of proof effectively separates the probabilistic and deterministic aspects of the argument through the introduction of an extended corrector. The statistical properties of the environment are used to prove the sublinearity of the large-scale averages of this corrector, which subsequently provides the starting point for a Campanato iteration. The latter is used to establish, almost surely, an intrinsic large-scale Hölder-regularity estimate for caloric functions.