Sur le comportement de la mesure invariante du procès de diffusion avec petit diffusion

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 $\varepsilon\to 0$.