  In this paper, we study limit behaviors of stationary measures of the
Fokker-Planck equations associated with a system of ordinary differential
equations perturbed by a class of multiplicative including additive white
noises. As the noises are vanishing, various results on the invariance and
concentration of the limit measures are obtained. In particular, we show that
if the noise perturbed systems admit a uniform Lyapunov function, then the
stationary measures form a relatively sequentially compact set whose
weak$^*$-limits are invariant measures of the unperturbed system concentrated
on its global attractor. In the case that the global attractor contains a
strong local attractor, we further show that there exists a family of
admissible multiplicative noises with respect to which all limit measures are
actually concentrated on the local attractor; and on the contrary, in the
presence of a strong local repeller in the global attractor, there exists a
family of admissible multiplicative noises with respect to which no limit
measure can be concentrated on the local repeller. Moreover, we show that if
there is a strongly repelling equilibrium in the global attractor, then limit
measures with respect to typical families of multiplicative noises are always
concentrated away from the equilibrium. As applications of these results, an
example of stochastic Hopf bifurcation is provided.
  Our study is closely related to the problem of noise stability of compact
invariant sets and invariant measures of the unperturbed system.
