Corrector equations and random operators
Consider a partial differential equation (PDE) with random coefficients as in engineering or applied physics: When combined with a spatial scale separation, the randomness and the differential operator interact to give rise to some effective behavior. The recent growing mathematical activity in this domain has led to a ``seemingly'' complete theory of stochastic homogenization  of linear elliptic operators. Central to this theory is the so-called corrector equation, a degenerate elliptic equation posed on the (infinite-dimensional) probability space. The context of linear elliptic operators yields the simplest such equation.  Time-dependent and/or nonlinear PDEs also involve corrector equations (or a family thereof), albeit with a significantly more complex structure. Their study and use to characterize the large-scale/time behavior of solutions of PDEs with random coefficients are at the heart of this project.  Whereas the relevance of corrector equations is clear in problems such as diffusion in random media, sedimentation of randomly placed particles in a fluid, or water waves on a rough bottom, it is less obvious for the long-time behavior of waves in disordered media. The latter is related to the spectrum of the associated random elliptic operator, the characterization of which still remains a largely open question today. We propose to relate the long-time behavior of waves to the properties of a family of corrector equations.     These corrector equations are widely unstudied and offer many analytical challenges. They constitute the first half of the project. Even in the ``well-understood'' setting of linear elliptic operators, this requires to revisit the corrector equation in the light of much weaker topologies than considered before.  The second half of the project aims at using correctors to establish the large-scale behavior of solutions as random objects. This may involve surprising quantities such as the recently introduced ``homogenization commutator"".