On the Behavior of the Invariant Measure of a Diffusion Process with Small Diffusion on a Circle

In this note we examine the behavior of the invariant measure $\mu _\varepsilon (v) = \int_v {p_\varepsilon } (x)dx$ of a Markov process, when the diffusion coefficient is a small parameter. In the case when the bounded dynamical system has an invariant measure with density $p_0 (x)$ we have shown that $\lim _{\varepsilon \to 0} p_\varepsilon (x) = p_0 (x)$. We have investigated the case when the bounded dynamical system has a stable position. Theorem 3 allows one to find the points in which the whole measure $\mu _\varepsilon (v)$ is concentrated as $\varepsilon \to 0$.