On the stochastic homogenization of fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media

We study homogenization for fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media from the qualitative and quantitative perspectives. Under suitable hypotheses, solutions to fully nonlinear uniformly parabolic equations in spatio-temporal media homogenize almost surely. In addition, we obtain a logarithmic rate of convergence for this homogenization in measure, assuming that the environment is strongly mixing with a prescribed logarithmic rate. A general methodology to study the stochastic homogenization and rates of convergence for stochastic homogenization of uniformly elliptic equations was introduced by Caffarelli, Souganidis, and Wang [1], and Caffarelli and Souganidis [2]. We extend their approach to fully nonlinear uniformly parabolic equations, developing a number of new arguments to handle the parabolic structure of the problem.