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