Page 192 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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OPERATOR SEMIGROUPS AND EVOLUTION EQUATIONS (MS-29)
generated by Markov processes which are obtained by different operations from some original
Markov processes. In this talk, we discuss Chernoff approximations for such operations as: a
random time change via an additive functional of a process, a subordination (i.e., a random time
change via an independent a.s. nondecreasing 1-dim. Lévy process), killing of a process upon
leaving a given domain, reflecting of a process. These results allow, in particular, to obtain
Chernoff approximations for subordinate diffusions on strar graphs and compact Riemannian
manifolds. Moreover, the constructed Chernoff approximations for evolution semigroups can
be used further to approximate solutions of some time-fractional evolution equations describing
anomalous diffusion (solutions of such equations do not posess the semigroup property and are
related to some non-Markov processes).
References
[1] Ya. A. Butko. The method of Chernoff approximation, to appear in “‘Semigroups of Oper-
ators: Theory and Applications SOTA 2018”, Springer Proceedings in Mathematics, 2020;
https://arxiv.org/pdf/1905.07309.pdf.
[2] Ya. A. Butko. Chernoff approximation for semigroups generated by killed Feller processes
and Feynman formulae for time-fractional Fokker–Planck–Kolmogorov equations. Fract.
Calc. Appl. Anal. 21 N 5 (2018), 35 pp.
[3] Ya. A. Butko. Chernoff approximation of subordinate semigroups. Stoch. Dyn. . Stoch.
Dyn., 18 N 3 (2018), 1850021, 19 pp., DOI: 10.1142/S0219493718500211.
[4] Ya. A. Butko, M. Grothaus and O.G. Smolyanov. Feynman formulae and phase space
Feynman path integrals for tau-quantization of some Lévy-Khintchine type Hamilton
functions. J. Math. Phys. 57 023508 (2016), 22 p.
[5] Ya. A. Butko. Description of quantum and classical dynamics via Feynman formulae.
Mathematical Results in Quantum Mechanics: Proceedings of the QMath12 Conference,
p.227-234. World Scientific, 2014. ISBN: 978-981-4618-13-7 (hardcover), ISBN: 978-
981-4618-15-1 (ebook).
[6] Ya. A. Butko, R.L. Schilling and O.G. Smolyanov. Lagrangian and Hamiltonian Feyn-
man formulae for some Feller semigroups and their perturbations, Inf. Dim. Anal. Quant.
Probab. Rel. Top., 15 N 3 (2012), 26 p.
[7] Ya. A. Butko, M. Grothaus and O.G. Smolyanov. Lagrangian Feynman formulae for sec-
ond order parabolic equations in bounded and unbounded domains, Inf. Dim. Anal. Quant.
Probab. Rel. Top. 13 N3 (2010), 377-392.
[8] Ya. A. Butko. Feynman formulas and functional integrals for diffusion with drift in a
domain on a manifold, Math. Notes 83 N3 (2008), 301–316.
190
generated by Markov processes which are obtained by different operations from some original
Markov processes. In this talk, we discuss Chernoff approximations for such operations as: a
random time change via an additive functional of a process, a subordination (i.e., a random time
change via an independent a.s. nondecreasing 1-dim. Lévy process), killing of a process upon
leaving a given domain, reflecting of a process. These results allow, in particular, to obtain
Chernoff approximations for subordinate diffusions on strar graphs and compact Riemannian
manifolds. Moreover, the constructed Chernoff approximations for evolution semigroups can
be used further to approximate solutions of some time-fractional evolution equations describing
anomalous diffusion (solutions of such equations do not posess the semigroup property and are
related to some non-Markov processes).
References
[1] Ya. A. Butko. The method of Chernoff approximation, to appear in “‘Semigroups of Oper-
ators: Theory and Applications SOTA 2018”, Springer Proceedings in Mathematics, 2020;
https://arxiv.org/pdf/1905.07309.pdf.
[2] Ya. A. Butko. Chernoff approximation for semigroups generated by killed Feller processes
and Feynman formulae for time-fractional Fokker–Planck–Kolmogorov equations. Fract.
Calc. Appl. Anal. 21 N 5 (2018), 35 pp.
[3] Ya. A. Butko. Chernoff approximation of subordinate semigroups. Stoch. Dyn. . Stoch.
Dyn., 18 N 3 (2018), 1850021, 19 pp., DOI: 10.1142/S0219493718500211.
[4] Ya. A. Butko, M. Grothaus and O.G. Smolyanov. Feynman formulae and phase space
Feynman path integrals for tau-quantization of some Lévy-Khintchine type Hamilton
functions. J. Math. Phys. 57 023508 (2016), 22 p.
[5] Ya. A. Butko. Description of quantum and classical dynamics via Feynman formulae.
Mathematical Results in Quantum Mechanics: Proceedings of the QMath12 Conference,
p.227-234. World Scientific, 2014. ISBN: 978-981-4618-13-7 (hardcover), ISBN: 978-
981-4618-15-1 (ebook).
[6] Ya. A. Butko, R.L. Schilling and O.G. Smolyanov. Lagrangian and Hamiltonian Feyn-
man formulae for some Feller semigroups and their perturbations, Inf. Dim. Anal. Quant.
Probab. Rel. Top., 15 N 3 (2012), 26 p.
[7] Ya. A. Butko, M. Grothaus and O.G. Smolyanov. Lagrangian Feynman formulae for sec-
ond order parabolic equations in bounded and unbounded domains, Inf. Dim. Anal. Quant.
Probab. Rel. Top. 13 N3 (2010), 377-392.
[8] Ya. A. Butko. Feynman formulas and functional integrals for diffusion with drift in a
domain on a manifold, Math. Notes 83 N3 (2008), 301–316.
190
