Semigroups of operators related to stochastic processes in an extension of the Gelfand-Shilov classification

Semigroups of operators corresponding to stochastic Levy processes are considered, and their connection with pseudo-differential $(\Psi D)$ operators is studied. It is shown that the semigroup generators are $\Psi D$-operators and operators with kernels from the space of slowly growing distributions. A classification of Cauchy problems is constructed for equations with operators from a special class of $\Psi D$-operators with polynomially bounded symbols. The constructed classification extends the Gelfand–Shilov classification for differential systems. In the extended classification, Cauchy problems with generators corresponding to Levy processes are well-posed in the sense of Petrovskii.