Stochastic Homogenization for Some Nonlinear Integro-Differential Equations

In this note we prove the stochastic homogenization for a large class of fully nonlinear elliptic integro-differential equations in stationary ergodic random environments. Such equations include, but are not limited to, Bellman equations and the Isaacs equations for the control and differential games of some pure jump processes in a random, rapidly varying environment. The translation invariant and non-random effective equation is identified, and the almost everywhere in ω, uniform in x convergence of the family solutions of the original equations is obtained. Even in the linear case of the equations contained herein the results appear to be new.