Levy processes.\
4. In sections 6-8 we investigate the structure and the properties of the quasi-potential $B$. In particular, we prove that the operator $B$ is compact and the following important inequality: $$\Phi(x,y){\geq}0$$ is true. From inequality (0.4) and Krein-Rutman theorem \[13\] we deduce that the operator $B$ has a positive eigenvalue $\lambda_{1}$, which is no less in modulus than every other eigenvalue of $B$.\
5. Section 9 contains formulas and estimations for the probability $p(t,\Delta)$ that a sample of the process $X_{\tau}$ remains inside the given domain $\Delta$, when $0{\leq}\tau{\leq}t$. Under certain conditions we have obtained the asymptotic formula $$p(t,\Delta)=e^{-t/\lambda_{1}}[c_{1}+o(1)],\quad t{\to}\infty.$$ 6. In sections 10-12 we separately consider a special case of the Levy processes, i.e., stable processes. We use the notation $p(t,a)=p(t,\Delta)$ when $\Delta=[-a,a].$ We consider the important case, where $a$ depends on $t$ and $a(t){\to}\infty$ when $t{\to}\infty.$ We compare the obtained results with well-known results (the iterated logarithm law, the results for the first hitting time, the results for the most visited sites problems). Further we introduce the notation $p(t,-b,a)=p(t,\Delta)$ when $\Delta=[-b,a].$ In case of the Wiener process we found the asymptotic behavior of $p(t,-b,a)$ when $b{\to}\infty.$ It is easy to see that $p(t,-\infty,a)$ coincides with the 