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The statistical counterpart of the formalism of hamiltonian systems withconvex dissipation arXiv:0810.1419 , arXiv:1408.3102 is a completely opensubject. Here are described a stochastic version of the SBEN principle and aLiouville type theorem which uses a minimal dissipation cost functional.

A stochastic version and a Liouville theorem for hamiltonian inclusions with convex dissipation