A system of ordinary differential equations, as a physical model, is often subject to white noise perturbations, either from its own uncertainty and complexity, or from an external source. The understanding of noise impacts on its essential dynamics is a fundamental dynamical issue of importance in both theory and applications. This talk will address this issue by investigating stationary measures of the noise-perturbed system. New existence and non-existence results of stationary measures will be presented. As noises vanish, concentration and limit behaviors of stationary measures will be described with particular attentions paying to the special role played by multiplicative noises. Connections to problems such as stochastic stability and stochastic bifurcations will also be discussed.

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