Average principles for forward-backward multivalued stochastic systems and homogenization for systems of nonlinear parabolic PDEs

This work concerns about forward-backward multivalued stochastic systems. First of all, we prove one average principle for general stochastic differential equations in the $L^{2p}$ ($p\ge 1$) sense. Moreover, for $p=1$ a convergence rate is presented. Then combining general stochastic differential equations with backward stochastic variation inequalities, we establish the other average principle for backward stochastic variation inequalities in the $L^2$ sense through a time discretization method. Finally, we apply our result to nonlinear parabolic partial differential equations and obtain the homogenization of them.