we have $(f,Sf){\ne}0$,when $||f||{\ne}0$. In particular the relation $(N_{1},SN_{1})=r{\ne}0$ is true.\
**Remark 8.2.** In view of (6.4) and (6.5) Proposition 8.1 is valid not only in the case of the symmetric segment $[-c,c]$ but in the general case $[-a,b]$ too.\

Long time behavior
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1\. In order to investigate the asymptotic behavior of $p(t,\Delta)$ when $t{\to}\infty$, we use the non-negativity of the kernel $\Phi(x,y)$. We apply the following Krein-Rutman theorem (see \[13\], section 6).\
**Theorem 9.1.** *If a linear compact operator $B$ leaving invariant a cone $K$, has a point of the spectrum different from zero , then it has a positive eigenvalue $\lambda_{1}$ not less in modulus than any other eigenvalues $\lambda_{k},\quad (k>1)$. To this eigenvalue $\lambda_{1}$ corresponds at least one eigenvector $g_{1}{\in}K, (Bg_{1}=\lambda_{1}g_{1})$ of the operator $B$ and at least one eigenvector $h_{1}{\in}K^{\star},
(B^{\star}h_{1}=\lambda_{1}h_{1})$ of the operator $B^{\star}$.*\
We remark that in our case the cone $K$ consists of non-negative functions $f(x){\in}L^{p}(\Delta)$. Hence we have $$g_{1}(x){\geq}0,\quad h_{1}(x){\geq}0.$$ We introduce the following normalizing condition $$(g_{1},h_{1})=\int_{\Delta}g_{1}(x)h_{1}(x)dx=1,$$ Let the interval $\Delta_{1}$ and the point $x_{0}$ be such that $$x_{0}{\in}\Delta_{1}{\in}\Delta.$$ Together with quantity $p(t,\Delta)$ we consider the expression $$p(x_{0},\Delta_{1},t,\Delta)=P(\underset{0{\leq}\tau{\leq}t}{(X_{\tau}{\in}\Delta)}{\bigcap}(X_{t}{\in}\Delta_{1})),$$where $x_{0}=X_{0}$. If the relations $x_{0}=0,\quad \Delta_{1}=\Delta $ are true, then $p(x_{0},\Delta_{1},t,\Delta)=p(t,\Delta).$ In this 