  This paper studies limit measures of stationary measures of stochastic
ordinary differential equations on the Euclidean space and tries to determine
which invariant measures of an unperturbed system will survive. Under the
assumption for SODEs to admit the Freidlin-Wentzell or Dembo-Zeitouni large
deviations principle with weaker compactness condition, we prove that limit
measures are concentrated away from repellers which are topologically
transitive, or equivalent classes, or admit Lebesgue measure zero. We also
preclude concentrations of limit measures on acyclic saddle or trap chains.
This illustrates that limit measures are concentrated on Liapunov stable
compact invariant sets. Applications are made to the Morse-Smale systems, the
Axiom A systems including structural stability systems and separated star
systems, the gradient or gradient-like systems, those systems possessing the
Poincare-Bendixson property with a finite number of limit sets to obtain that
limit measures live on Liapunov stable critical elements, Liapunov stable basic
sets, Liapunov stable equilibria and Liapunov stable limit sets including
equilibria, limit cycles and saddle or trap cycles, respectively. A number of
nontrivial examples admitting a unique limit measure are provided, which
include monostable, multistable systems and those possessing infinite
equivalent classes.
