We consider white noise perturbations of a system of ordinary differential equations. By relaxing the notion of Lyapunov functions associated with the stationary Fokker-Planck equations, new existence and non-existence results of stationary measures in a general domain including the entire space will be presented for both non-degenerate and degenerate noises. Limiting behaviors of stationary measures will be discussed along with applications to problems of stochastic stability and bifurcations.