We exploit limiting measures of stationary measures of stochastic ordinary differential equations. Using the Freidlin-Wentzell large deviations principle, we prove that limiting measures are concentrated away from repellers which are topologically transitive, or equivalent classes, or admit Lebesgue measure zero. We also preclude concentrations of limiting measures on acyclic saddle or trap chains. This implies that any limiting measure will concentrated on Liapunov stable compact invariant sets if unperturbed ordinary differential equations are dissipative. Applications are made to the Morse-Smale systems, the Axiom A systems including structural stability systems and separated start systems, the gradient or gradient-like systems, those systems possessing the Poincaré-Bendixson property with a finite number of limit sets to obtain that limiting measures live on Liapunov stable critical elements, Liapunov stable basic sets, Liapunov stable equilibria, Liapunov stable limit sets including saddle or trap cycles, respectively. Together these results with the Laplace method, we give the exact concentration of limiting measure of quasipotential systems, which is the support of stochastically stable invariant measure.

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