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In our previous paper (ArXiv:1306.1492) we have proved that a representationof the infinitesimal generators $L$ for Levy processes $X_t$ can be writtendown in a convolution type form. For the case of non-summable Levy measures weconstructed the quasi-potential operators $B$ and investigated the long timebehavior of $X_t$. In the present paper we consider Levy processes $X_t$ withsummable Levy measures. In this case the form of the quasi-potential operators$B$ essentially differs from the form in the case of non-summable Levymeasures. We use this new form in order to study the long time behavior of$X_t$ for the case of summable Levy measures.

Levy processes with summable Levy measures, long time behavior