$$\rho(x,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-ixz}\mu(z,t)dz,\quad t>0$$ Now we introduce the sequence of functions $$Q_{n+1}(x,t)=\int_{0}^{t}\int_{-\infty}^{\infty}Q_{0}(x-\xi,t-\tau)V(\xi)Q_{n}(\xi,\tau)d{\xi}d\tau,$$ where the function $V(x)$ is defined by relations $V(x)=1$ when $x{\notin}\Delta$ and $V(x)=0$ when $x{\in}\Delta.$ We use the notation $$Q_{0}(x,t)=\rho(x,t).$$ For Levy processes the following relation $$Q_{0}(x,t)=\int_{-\infty}^{\infty}Q_{0}(x-\xi,t-\tau)Q_{0}(\xi,\tau)d{\xi}$$ is true. Using (5.3) and (5.5) we have $$0{\leq}Q_{n}(x,t){\leq}t^{n}Q_{0}(x,t)/n!.$$ Hence the series $$Q(x,t,u)=\sum_{n-0}^{\infty}(-1)^{n}u^{n}Q_{n}(x,t)$$ converges. The probabilistic meaning of $Q(x,t,u)$ is defined by the relation (see \[12\],Ch.4) $$E\{\mathrm{exp}[-u\int_{0}^{t}V(X_{\tau})d\tau],c_{1}<X_{t}<c_{2}\}=
\int_{c_{1}}^{c_{2}}Q(x,t,u)dx.$$The inequality $V(x){\geq}0$ and relation (5.8) imply that the function $Q(x,t,u)$ monotonically decreases with respect to the variable “u” and the formulas $$0{\leq}Q(x,t,u){\leq}Q(x,t,0)=Q_{0}(x,t)=\rho(x,t)$$ are true. In view of (5.2) and (5.9) the Laplace transform $$\psi(x,s,u)=\int_{0}^{\infty}e^{-st}Q(x,t,u)dt,\quad s>0.$$ has the meaning. According to (5.3) the function $Q(x,t,u)$ is the solution of the equation $$Q(x,t,u)+u\int_{0}^{t}\int_{-\infty}^{\infty}\rho(x-\xi,t-\tau)V(\xi)Q(\xi,\tau,u)d{\xi}d\tau=
\rho(x,t)$$ Taking from both parts of (5.11) the Laplace transform and bearing in mind (5.10) we obtain $$\psi(x,s,u)+u\int_{-\infty}^{\infty}V(\xi)R(x-\xi,s)\psi(\xi,s,u)d\xi=R(x,s),$$ where $$R(x,s)=\int_{0}^{\infty}e^{-st}\rho(x,t)dt.$$Multiplying both parts of relation (5.12) by $\mathrm{exp}(ixp)$ and integrating them with respect to $x \quad (-\infty<x<\infty)$ we have $$\int_{-\infty}^{\infty}\psi(x,s,u)e^{ixp}[s+\lambda(p)+uV(x)]dx=1.$$ Here we use relations (1.1), (5.2) and (5.13). Now we introduce the function $$h(p)=\frac{1}{2\pi}\int_{\Delta}e^{-ixp}f(x)dx,$$ where the function $f(x)$ belongs to $C_{\Delta}$. Multiplying both parts of (5.14) by $h(p)$ and integrating them with respect to $p\quad  (-\infty<p<\infty)$ we deduce the equality $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\psi(x,s,u)e^{ixp}[s+\lambda(p)]h(p)dxdp=f(0).$$ We have used the 