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In the present paper we show that the Levy-Ito representation of theinfinitesimal generator $L$ for Levy processes $X_t$ can be written in aconvolution-type form. Using the obtained convolution form we have constructedthe quasi-potential operator $B$. We denote by $p(t,\Delta)$ the probabilitythat a sample of the process $X_t$ remains inside the domain $\Delta$ for$0{\leq}\tau{\leq}t$ (ruin problem). With the help of the operator $B$ we finda new formula for $p(t,\Delta)$. This formula allows us to obtain long timebehavior of $p(t,\Delta)$.

Levy processes: long time behavior and convolution-type form of the Ito representation of the infinitesimal generator