The first boundary exit point of a diffusion process with small diffusion

A diffusion process with a drift $b(x)$ and small diffusion sooner or later reaches the boundary of a region containing a stable position of equilibrium of the dynamical system $\dot x_t=b(x_t)$. The first exit point belongs to a small part of the boundary (see [1, 2]). An estimate of the order of the size of this part is given.