Geodesic
Semigroups of operators related to stochastic processes in an extension of the Gelfand-Shilov classification
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 4, pp. 74-87

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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.
Keywords: Levy process, transition probability, semigroup of operators, pseudo-differential operator, Levy–Khintchine formula. 
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     author = {I. V. Mel'nikova and V. A. Bovkun},
     title = {Semigroups of operators related to stochastic processes in an extension of the {Gelfand-Shilov} classification},
     journal = {Trudy Instituta matematiki i mehaniki},
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     url = {http://geodesic.mathdoc.fr/item/TIMM_2021_27_4_a5/}
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I. V. Mel'nikova; V. A. Bovkun. Semigroups of operators related to stochastic processes in an extension of the Gelfand-Shilov classification. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 4, pp. 74-87. http://geodesic.mathdoc.fr/item/TIMM_2021_27_4_a5/