Some results concerning small stochastic perturbations of dynamical systems.

Let
$$ dx_t^\varepsilon=\varepsilon dw_t+b(x_t^\varepsilon)\,dt $$
and $p^\varepsilon(t,x,y)$ be the transition probability density of $x_t^\varepsilon$. In section 1, we find an exact asymptotics of $p^\varepsilon(t,x,y)$ as $\varepsilon\to0$. Section 2 is devoted to investigation of the behaviour of $\mathbf P_x^\varepsilon\{x_\tau^\varepsilon\in\Delta,\ \tau\le T\}$ as $\varepsilon\to0$, where $\Delta$ is an open subset of the boundary $\Gamma$ of a bounded domain $G$ and $\tau$ is first exit time from $G$ $(x\in G)$.
Let $b(x)=Bx$, where $B$ is a matrix the eigenvalues of which have negative real parts. In this case we get an exact asymptotics of $\mathbf P_x^\varepsilon\{x_\tau^\varepsilon\in\Delta\}$ as $\varepsilon\to0$.