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The existence of global martingale weak solution for the 2D and 3D stochasticCahn-Hilliard-Navier-Stokes equations driven by multiplicative noise in asmooth bounded domain is established. In particular, the system is suppliedwith the dynamic boundary condition which accounts for the interaction betweenthe fluid components and the rigid walls. The proof is completed by athree-level approximate scheme combining a fixed point argument and thestochastic compactness argument, overcoming challenges from strongnonlinearity, dynamic boundary and random effect. Then, we prove the existenceof an almost surely Markov selection to the associated martingale problemfollowing the abstract framework established by F. Flandoli and M. Romito.

Stochastic Cahn-Hilliard-Navier-Stokes equations with the dynamic boundary: Martingale weak solution, Markov selection