I will consider deterministic and stochastic perturbations of dynamical systems and stochastic processes with conservation laws. Slow evolution of first integrals defines the long-time behavior of the perturbed system. If the collection of first integrals is "complete enough", its evolution converges, in an appropriate time scale, to a Markov process on corresponding phase space. Even pure deterministic perturbations of deterministic systems may lead to random long- time evolution due to instabilities in the non-perturbed system.
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