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In this paper, we establish the second Bogolyubov theorem and globalaveraging principle for stochastic partial differential equations (in short,SPDEs) with monotone coefficients. Firstly, we prove that there exists a unique$L^{2}$-bounded solution to SPDEs with monotone coefficients and this boundedsolution is globally asymptotically stable in square-mean sense. Then we showthat the $L^{2}$-bounded solution possesses the same recurrent properties (e.g.periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoffrecurrent, Levitan almost periodic, etc.) in distribution sense as thecoefficients. Thirdly, we prove that the recurrent solution of the originalequation converges to the stationary solution of averaged equation under thecompact-open topology as the time scale goes to zero--in other words, thereexists a unique recurrent solution to the original equation in a neighborhoodof the stationary solution of averaged equation when the time scale is small.Finally, we establish the global averaging principle in weak sense, i.e. weshow that the attractor of original system tends to that of the averagedequation in probability measure space as the time scale goes to zero. Forillustration of our results, we give two applications, including stochasticreaction diffusion equations and stochastic generalized porous media equations.

The second Bogolyubov theorem and global averaging principle for SPDEs with monotone coefficients