On the asymptotic behaviour of the first eigenvalue of a second-order differential operator with small parameter by the higher derivatives

An application of results concerning Markow processes to investigation of eigenvalues of linear operators is given.
Let $L^h=\sum b^i(x)\partial/\partial x^i+(h/2)\sum a^{ij}(x)\partial^2/\partial x^i\partial x^j$, for each $h>0$, be an elliptic operator in a bounded domain $D\subset R^r$; $\lambda_1(h)$ be the first (i.e., minimal) eigenvalue of the operator $-L^h$ with zero boundary condition on $\partial D$. It was shown in [2], [3] that, if in $D$ there exists a finite number of compacts containing stable $\omega$-limiting sets of the dynamical system $\dot x_t=b(x_t)$, then $\lambda_1(h)$ tends to 0 with an exponential rate when $h\downarrow0$. In this paper, we show that, if all solutions of $\dot x_t=b(x_t)$ sooner or later leave $D\bigcup\partial D$, then $\lambda_1(h)=c_1h^{-1}+o(h^{-1})$; a formula for the constant $c_1$ is given. The proof, as well as in [2], uses the diffusion process $(x_t^h,\mathbf P_x^h)$ corresponding to $L^h$ and the exit time $\tau^h$ for $D$ and the theorems of [1] concerning probabilities of certain nearly improbable events.