On small random perturbations of dynamical systems

In this paper we study the effect on a dynamical system $\dot x_t=b(x_t)$ of small random perturbations of the type of white noise:
$$ \dot x_t^\varepsilon=b^\varepsilon(x_t^\varepsilon) +\varepsilon \sigma (x_t^\varepsilon)\bar\xi_t, $$
where $\xi_t$ is the $r$-dimensional Wiener process and $b^\varepsilon(x)\to b(x)$ as $\varepsilon\to 0$. We are mainly concerned with the effect of these perturbations on long time-intervals that increase with the decreasing $\varepsilon$. We discuss two problems: the first is the behaviour of the invariant measure $\mu^\varepsilon$ of the process $x_t^\varepsilon$ as $\varepsilon\to 0$, and the second is the distribution of the position of a trajectory at the first time of its exit from a compact domain. An important role is played in these problems by an estimate of the probability for a trajectory of $x_t^\varepsilon$ not to deviate from a smooth function $\varphi_t$ by more than $\delta$ during the time $[0, T]$. It turns out that the main term of this probability for sma $\varepsilon$ and $\delta$ has the form $\exp\bigl\{-\frac{1}{2\varepsilon^2}I(\varphi)\bigr\}$ where $I(\varphi)$, is a certain non-negative functional of $\varphi_t$. A function $V(x,y)$, the minimum o $I(\varphi)$ over the set of all functions connecting $x$ and $y$, is involved in the answers to both the problems.
By means of $V(x,y)$ we introduce an independent of perturbations relation of equivalence in the phase-space. We show, under certain assumption, at what point of the phase-space the invariant measure concentrates in the limit.
In both the problems we approximate the process in question by a certain Markov chain; the answers depend on the behaviour of $V(x,y)$ on graphs that are associated with this chain.
Let us remark that the second problem is closely related to the behaviour of the solution of a Dirichlet problem with a small parameter at the highest derivatives.