Abstract
We develop the idea of non-Markovian CTRW (continuous time random walk) approximation to the evolution of interacting particle systems, which leads to a general class of fractional kinetic measure-valued evolutions with variable order. We prove the well-posedness of the resulting new equations and present a probabilistic formula for their solutions. Though our method are quite general, for simplicity we treat in detail only the fractional versions of the interacting diffusions. The paper can be considered as a development of the ideas from the works of Belavkin and Maslov devoted to Markovian (quantum and classical) systems of interacting particles.
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References
M. A. Leontovich, “Main equations of the kinetic theory from the point of view of random processes,” Zh. Exper. Teoret. Fiz. 5, 211–231 (1935).
V. P. Belavkin and V. N. Kolokoltsov, “On general kinetic equation for many particle systems with interaction, fragmentation and coagulation,” R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 459 (2031), 727–748 (2003).
V. P. Belavkin and V. P. Maslov, “The Method of uniphormization in the theory of nonlinear Hamiltonian systems of the type of Vlasov and Hartry,” Theoret. Math. Phys.] 33 (1), 852–862 (1977).
M. Kac, Probability and Related Topics in Physical Science (Intersci. Publ., London, 1959).
V. P. Maslov and S. É. Tariverdiev, “Asymptotics of the Kolmogorov–Feller equation for a system of a large number of particles,” J. Math. Sci. (New York) 23 (5), 2553–2579 (1983).
D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Series on Complexity, Nonlinearity and Chaos, Vol. 5: Fractional Calculus. Models and Numerical Methods (World Sci. Publ., Singapore, 2017).
V. Kiryakova, Pitman Research Notes in Mathematics Series, Vol. 301: Generalized Fractional Calculus and Applications (Longman Sci., New York, 1994).
I. Podlubny, Mathematics in Science and Engineering. Vol. 198: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (Academic Press, San Diego, CA, 1999).
V. N. Kolokoltsov, CTRW Approximations for Fractional Equations With Variable Order, arXiv: 2203.04897 (2022).
V. N. Kolokoltsov, Cambridge Tracts in Mathematics, Vol. 182: Nonlinear Markov Processes and Kinetic Equations (Cambridge, Cambridge Univ. Press, 2010).
V. N. Kolokoltsov and M. Troeva, “A new approach to fractional kinetic evolutions,” Fractal and Fractional 6 (2), 49 (2022).
V. N. Kolokoltsov, “Generalized continuous-time random walks (CTRW), subordination by Hitting times and fractional dynamics,” Theory Probab. Appl. 53 (4), 594–609 (2008).
A. N. Kochubei and Y. Kondratiev, “Fractional kinetic hierarchies and intermittency,” Kinet. Relat. Models 10 (3), 725–740 (2017).
V. N. Kolokoltsov and O. A. Malafeyev, Many Agent Games in Socio-Economic Systems. Corruption, Inspection, Coalition Building, Network Growth, Security (Springer, Cham, 2019).
V. N. Kolokoltsov, Differential Equations on Measures and Functional Spaces (Birkhäuser, Cham, 2019).
M. M. Meerschaert and H.-P. Scheffler, Limit Distributions for Sums of Independent Random Vectors. Heavy Tails in Theory and Practice (John Wiley and Son, New York, 2001).
V. N. Kolokoltsov, De Gruyter Studies in Mathematics, Vol. 38: Markov Processes, Semigroups and Generators (Walter de Gruyter, Berlin, 2011).
O. Kallenberg, Foundations of Modern Probability (Springer, New York, 2002).
B. Goldys, M. Nendel, and M. Röckner, Operator Semigroups in the Mixed Topology and the Infinitesimal Description of Markov Processes, arXiv: 2204.07484v1 (2022).
Acknowledgments
The authors are grateful to the referee for reading the article and providing a number of useful comments.
Funding
The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Fractional Differential Equations January–April 2022, where work on this paper was undertaken. The first author is grateful to the Simons foundation for the support of his residence at INI in Cambridge during the programme Fractional Differential Equations, January–April 2022. The work of V. N. Kolokoltsov (Secs. 1–5) was supported by the Russian Science Foundation under grant 20-11-20119), the work of M. S. Troeva (Secs. 6–8) was supported by the Ministry of Science and Higher Education of the Russian Federation (grant no. FSRG-2020-0006).
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Translated from Matematicheskie Zametki, 2022, Vol. 112, pp. 567–585 https://doi.org/10.4213/mzm13584.
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Kolokoltsov, V.N., Troeva, M.S. Fractional Kinetic Equations. Math Notes 112, 561–575 (2022). https://doi.org/10.1134/S0001434622090255
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DOI: https://doi.org/10.1134/S0001434622090255