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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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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/

[1] Trev F., Vvedenie v teoriyu psevdodifferentsialnykh operatorov i integralnykh operatorov Fure, v 2 t., v. 1, Psevdodifferentsialnye operatory, Mir, M., 1984, 360 pp.

[2] Applebaum D., Levy processes and stochastic calculus, Cambridge Studies in Advanced Math., 116, Cambridge University Press, Cambridge, 2009, 492 pp. | DOI | Zbl

[3] Gelfand I.M., Shilov G.E., Obobschennye funktsii, v. Vyp. 3, Nekotorye voprosy teorii differentsialnykh uravnenii, Fizmatgiz, M., 1958, 276 pp.

[4] Böttcher B., Schilling R., Wang J., Levy matters III. Levy-type processes: construction, approximation and sample path properties, Springer, Cham; Heidelberg; NY; Dordrecht; London, 2013, 199 pp. | DOI | Zbl

[5] Bulinskii A.V., Shiryaev A.N., Teoriya sluchainykh protsessov, Fizmatlit, M., 2005, 408 pp.

[6] Sato K.-I., Levy processes and infinitely divisible distributions, Cambridge University Press, Cambridge, 2013, 536 pp. | Zbl

[7] Balakrishnan A.V., Prikladnoi funktsionalnyi analiz, Nauka, M., 1980, 383 pp.

[8] Kolokoltsov V.N., Markov processes, semigroups and generators, De Gruyter Studies in Math., 38, De Gruyter, Berlin; NY, 2011, 430 pp. | DOI | Zbl

[9] Khermander L., Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, v. 1, Mir, M., 1986, 464 pp.

[10] Reed M., Simon B., Methods of modern mathematical physics, v. 4, Analysis of operators, Acad. Press, NY; London, 1978, 325 pp. | Zbl

[11] Jacob N., Pseudo-differential operators and Markov processes, v. 1, Imperial College Press, London, 2001, 493 pp. | Zbl

[12] Melnikova I.V., Stochastic Cauchy problems in infinite dimensions. Regularized and generalized solutions, CRC Press, NY, 2016, 306 pp. | DOI | Zbl

[13] Anufrieva U.A., Melnikova I.V., “Osobennosti i regulyarizatsiya nekorrektnykh zadach Koshi s differentsialnymi operatorami”, Sovremennaya matematika. Fundamentalnye napravleniya, 14 (2005), 3–156 | Zbl

[14] Reed M., Simon B., Methods of modern mathematical physics, v. 2, Fourier analysis, self-adjointness, Acad. Press, NY; London, 1975, 384 pp. | Zbl