  In this article we consider the Levy processes and the corresponding
semigroup. We represent the generator of this semigroup in a convolution form.
Using the obtained convolution form and the theory of integral equations we
investigate the properties of a wide class of Levy processes (potential,
quasi-potential, the probability of the Levy process remaining within the given
domain, long time behavior, stable processes). We analyze in detail a number of
concrete examples of the Levy processes (stable processes, the variance damped
Levy processes, the variance gamma processes, the normal Gaussian process, the
Meixner process, the compound Poisson process).
