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A general approach to a broad class of asymptotic problems related tolong-time influence of small perturbations, of both deterministic andstochastic type, is presented in the paper. The main characteristic of thisinfluence is a limiting motion on the simplex of invariant probability measuresof the non-perturbed system in an appropriate time scale. The main tools we usein the paper are limit theorems for large deviations, modified averagingprinciple, and diffusion approximation.

Long Time Influence of Small Perturbations and Motion on the Simplex of Invariant Probability Measures