I.V. Melnikova, V.A. Bovkun. Semigroups of operators related to stochastic processes in an extension of the Gelfand–Shilov classification ... P. 74-87

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

Received February 27, 2021

Revised September 1, 2021

Accepted September 6, 2021

Irina V. Melnikova, Dr. Phys.-Math. Sci., Prof., Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: Irina.Melnikova@urfu.ru

Vadim Andreevich Bovkun, Cand. Sci. (Phys.-Math.), Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: Vadim.Bovkun@urfu.ru

REFERENCES

1.   Treves F. Introduction to pseudodifferential and Fourier integral operators. Vol. 1: Pseudodifferential operators. Ser. University Series in Mathematics, NY: Springer US, 1980, 299 p.
doi: 10.1007/978-1-4684-8780-0 . Translated to Russian under the title Vvedenie v teoriyu psevdodifferentsial’nykh operatorov i integral’nykh operatorov Fur’e, vol. 1: Psevdodifferentsial’nye operatory, Moscow: Mir Publ, 1984, 360 p.

2.   Applebaum D. Levy processes and stochastic calculus. Ser. Cambridge Studies in Advanced Math., vol. 116. Cambridge: Cambridge University Press, 2009, 492 p. doi: 10.1017/CBO9780511809781 

3.   Gel’fand I.M., Shilov G.E. Generalized functions, vol. 3: Theory of differential equations. Providence: AMS Chelsea Publ., 1967, 222 p. ISBN: 978-1-4704-2661-3 . Original Russian text published in Gel’fand I.M., Shilov G.E. Obobshchennye funktsii. Vypusk 3: Nekotorye voprosy teorii differentsial’nykh uravnenii, Moscow: Fizmatgiz Publ., 1958, 276 p.

4.   Böttcher B., Schilling R., Wang J. Levy matters III. Levy-type processes: construction, approximation and sample path properties. Cham; Heidelberg; NY; Dordrecht; London: Springer, 2013, 199 p. doi: 10.1007/978-3-319-02684-8 

5.   Bulinskii A.V., Shiryaev A.N. Teoriya sluchainykh protsessov [Theory of stochastic processes]. Moscow: Fizmatlit Publ., 2005, 408 p. ISBN: 5-9221-0335-0 .

6.   Sato K.-I. Levy processes and infinitely divisible distributions. Cambridge: Cambridge University Press, 2013, 536 p. ISBN: 9781107656499 .

7.   Balakrishnan A.V. Applied functional analysis. NY: Springer-Verlag, 1981, 373 p. ISBN: 0387905278 . Original Russian text published in Balakrishnan A.V. Prikladnoi funktsional’nyi analiz, Moscow: Nauka Publ., 1980, 383 p.

8.   Kolokoltsov V.N. Markov processes, semigroups and generators. Ser. De Gruyter Studies in Math., vol. 38, Berlin; NY: De Gruyter, 2011, 430 p. doi: 10.1515/9783110250114 .

9.   H$\ddot{\mathrm{o}}$rmander L. The analysis of linear partial differential operators I. Berlin; Heidelberg: Springer-Verlag, 2003, 440 p. doi: 10.1007/978-3-642-61497-2 . Translated to Russian under the title Analiz lineinykh differentsial’nykh operatorov s chastnymi proizvodnymi. T. 1, Moscow: Mir Publ., 1986, 464 p.

10.   Reed M., Simon B. Methods of modern mathematical physics. Vol. 4: Analysis of operators. NY; London: Acad. Press, 1978, 325 p. ISBN: 9780080570457 .

11.   Jacob N. Pseudo-differential operators and Markov processes. Vol. 1. London: Imperial College Press, 2001, 493 p. ISBN: 9781783261345 .

12.   Melnikova I.V. Stochastic Cauchy problems in infinite dimensions. Regularized and generalized solutions. NY: CRC Press, 2016, 306 p. doi: 10.1201/9781315372631 

13.   Anufrieva U.A., Mel’nikova I.V. Peculiarities and regularization of ill-posed Cauchy problems with differential operators. J. Math. Sci., 2008, vol. 148, no. 4, pp. 481–632. doi: 10.1007/s10958-008-0012-5 

14.   Reed M., Simon B. Methods of modern mathematical physics. Vol. 2 : Fourier analysis, self-adjointness. NY; London: Acad. Press, 1975, 384 p. ISBN: 0125850026 .

Cite this article as: 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 Mekhaniki UrO RAN, 2021, vol. 27, no. 4, pp. 74–87.