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

摘要

This thesis is devoted to the study of homogenization for fully nonlinear uniformly parabolic equations in stationary ergodic spatio-temporal media. We examine the subject from both the qualitative and quantitative perspective. Under suitable hypotheses, we present multiple arguments to show that solutions to fully nonlinear uniformly parabolic equations in stationary ergodic media homogenize almost surely. We extend the methodology introduced by Caffarelli, Souganidis, and Wang as well as the approach of Armstrong and Smart to the parabolic setting. 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. We follow the strategy introduced by Caffarelli and Souganidis modified for the parabolic setting, and we develop a number of new arguments to handle the parabolic structure of the problem. In particular, we establish a quantitative interior regularity result for nonnegative supersolutions of fully nonlinear uniformly parabolic equations. The result may be interpreted as a nonlinear, quantitative version of a growth lemma established by Krylov and Safonov for nonnegative supersolutions of linear uniformly parabolic equations in nondivergence form.