Outleading sequences and continuous semi-Markov processes on the line.

A family $(g_\Delta, h_\Delta)$ ($\Delta$ is an interval) of desisions of the differential equation $f''+A(x)f'+B(x)f=0$ is considered, where $A$, $B$ are continuous functions and boundary conditions are as follows: for all $\Delta=(a, b), g_\Delta(a)=h_\Delta(b)=1, g_\Delta(b)=h_\Delta(a)=0$. Let $f_\Delta(B|x)=g_\Delta(x)\mathbb I(B|a)+h_\Delta(x)\mathbb I(B|b)$ if $x\in\Delta$ and $f_\Delta(B|x)=\mathbb I(B|x)$ if $x\notin\Delta,$ and $M_\Delta=\max_{x\in\Delta}|A(x)|$, $m_\Delta=-\max_{x\in\Delta}B(x)>0$. The following theorem is proved: if $M_{(-r, r)}/rm_{(-r, r)}\to0$ $(r\to\infty)$ then for each outleading sequence $(\Delta_1, \Delta_2,\dots)$ $(\forall x\in\mathbb R)$ $f_{(\Delta_1,\dots,\Delta_n)}(B|x)\to0$ $(n\to\infty)$ where $f_{(\Delta_1,\dots,\Delta_n)}(B|x)=\int_{\mathbb R}f_{\Delta_1}(dx_1|x)f_{(\Delta_2,\dots,\Delta_n)}(B|x_1)$ $(n\geqslant2, B\in\mathcal B(\mathbb R))$ is an iterated kernel. She sequence $(\Delta_1, \Delta_2,\dots)$ is called outleading one if $(\forall\xi\in\mathcal D(\mathbb R))$ $\tau_{(\Delta_1,\dots,\Delta_n)}\xi\to0$ $(n\to\infty)$ where $\tau_\Delta\xi=\inf\{t\geqslant0, \xi(t)\notin\Delta\}, \tau_{(\Delta_1,\dots,\Delta_n)}=\tau_{\Delta_1}+\tau_{(\Delta_2,\dots,\Delta_n)}\circ\theta_{\tau_{\Delta_1}}$, $\theta_\tau$ is a shift operator. This theorem is applied to prove the existence of a semi-Marcov process with the transition function to satisfy to defferential equation of the given form.