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We continue our study of the well-posedness theory of a one-parameter familyof coupled KdV-type systems in the periodic setting. When the value of acoupling parameter \alpha \in (0, 4) \setminus 1, we show that the Gibbsmeasure is invariant under the flow and the system is globally well-posedalmost surely on the statistical ensemble, provided that certain Diophantineconditions are satisfied.

Invariant Gibbs Measures and a.s. Global Well-Posedness for Coupled KdV Systems