From random times to fractional kinetics (original) (raw)

Мiждисциплiнарнi дослiдження складних систем

Abstract

In this paper we study the effect of the subordination by a general random time-change to the solution of a model on spatial ecology in terms of its evolution density. In particular on traveling waves for a non-local spatial logistic equation. We study the Cesaro limit of the subordinated dynamics in a number of particular cases related to the considered fractional derivative making use of the Karamata-Tauberian theorem.

Loading...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (51)

  1. T. M. Atanackovic, S. Pilipovic, and D. Zorica. Time distributed- order diffusion-wave equation. I., II. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, volume 465, pages 1869-1891, 1893-1917. The Royal Society, 2009.
  2. E. G. Bazhlekova. Subordination principle for fractional evolution equations. Fract. Calc. Appl. Anal., 3(3):213-230, 2000.
  3. E. G. Bazhlekova. Fractional Evolution Equations in Banach Spaces. PhD thesis, University of Eindhoven, 2001.
  4. E. Bazhlekova. Subordination principle for a class of fractional order differential equations. Mathematics, 3(2):412-427, 2015.
  5. N. H. Bingham. Limit theorems for occupation times of Markov processes. Z. Wahrsch. verw. Gebiete, 17:1-22, 1971.
  6. N. H. Bingham, C. M. Goldie, and J. L. Teugels. Regular varia- tion, volume 27 of Encyclopedia of Mathematics and its Applica- tions. Cambridge University Press, Cambridge, 1987.
  7. S. Bochner. Harmonic Analysis and the Theory of Probability. Univ. California Press, Berkeley, 1955.
  8. N. N. Bogoliubov. Problems of a dynamical theory in statistical physics. Gostekhisdat, Moskau, 1946. In Russian. English trans- lation in J. de Boer and G. E. Uhlenbeck, editors, Studies in Statistical Mechanics, volume 1, pages 1-118, Amsterdam, 1962. North-Holland.
  9. B. Bolker and S. W. Pacala. Using moment equations to under- stand stochastically driven spatial pattern formation in ecological systems. Theor. Popul. Biol., 52(3):179-197, 1997.
  10. L. Bondesson, G. K. Kristiansen, and F. W. Steutel. Infinite divisibility of random variables and their integer parts. Statist. Probab. Lett., 28:271-278, 1996.
  11. Zhen-Qing Chen, Panki Kim, Takashi Kumagai, and Jian Wang, Heat kernel estimates for time fractional equations. Forum Math., published online 2018-02-16.
  12. J. L. Da Silva, Y. G. Kondratiev, and P. Tkachov. Fractional kinetics in a spatial ecology model. Meth. Funct. Anal. Topology, 24(3):275-287,2018.
  13. V. Daftardar-Gejji and S. Bhalekar. Boundary value problems for multi-term fractional differential equations. J. Math. Anal. Appl., 345(2):754-765, 2008.
  14. S. D. Eidelman, S. D. Ivasyshen, and A. N. Kochubei. Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, volume 152. Springer Science & Business Media, 2004.
  15. D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Individual based model with competition in spatial ecology. SIAM J. Math. Anal., 41(1):297-317, 2009.
  16. D. L. Finkelshtein, Y. G. Kondratiev, and O. Kutoviy. Vlasov scaling for stochastic dynamics of continuous systems. J. Stat. Phys., 141(1):158-178, October 2010.
  17. D. Finkelshtein, Y. G. Kondratiev, and O. Kutoviy. Semigroup approach to birth-and-death stochastic dynamics in continuum. J. Funct. Anal., 262(3):1274-1308, 2012.
  18. D. Finkelshtein, Y. G. Kondratiev, Y. Kozitsky, and O. Ku- toviy. The statistical dynamics of a spatial logistic model and the related kinetic equation. Math. Models Methods Appl. Sci., 25(02):343-370, 2015.
  19. D. L. Finkelshtein, Y. G. Kondratiev, and P. Tkachov. Doubly nonlocal Fisher-KPP equation: Existence and properties of trav- eling waves. ArXiv:1804.10258v1, April 2018.
  20. N. Fournier and S. Méléard. A microscopic probabilistic descrip- tion of a locally regulated population and macroscopic approxi- mations. Ann. Appl. Probab., pages 1880-1919, 2004.
  21. R. Gorenflo, A. A. Kilbas, F. Mainardi, and V. R. Sergei. Mittag- Leffler Functions, Related Topics and Applications. Springer, 2014.
  22. G. Gripenberg. Volterra integro-differential equations with accre- tive nonlinearity. J. Differential Equations, 60(1):57-79, 1985.
  23. R. Gorenflo and S. Umarov. Cauchy and nonlocal multi-point problems for distributed order pseudo-differential equations, Part one. Z. Anal. Anwend., 24(3):449-466, 2005.
  24. A. Hanyga. Anomalous diffusion without scale invariance. J. Phys. A: Mat. Theor., 40(21):5551, 2007.
  25. M. J. Karamata. Sur un mode de croissance régulière. Théorèmes fondamentaux. Bull. Soc. Math. France, 61:55-62, 1933.
  26. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory and Applications of Fractional Differential Equations, volume 204 of North-Holland Mathematics Studies. Elsevier Science B.V., Am- sterdam, 2006.
  27. A. N. Kochubei. Distributed-order calculus: An operator- theoretic interpretation. Ukrainian Math. J., 60(4):551-562, 2008.
  28. A. N. Kochubei. Distributed order calculus and equations of ultraslow diffusion. J. Math. Anal. Appl., 340(1):252-281, 2008.
  29. A. N. Kochubei. General fractional calculus, evolution equa- tions, and renewal processes. Integral Equations Operator Theory, 71(4):583-600, October 2011.
  30. A. N. Kochubei and Y. G. Kondratiev. Fractional kinetic hier- archies and intermittency. Kinet. Relat. Models 10 (2017), no. 3, 725-740.
  31. V. N. Kolokoltsov. Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics. Theory Probab. Appl., 53(4), 594-609, 2009.
  32. V. N. Kolokoltsov. Markov Processes, Semigroups and generators, volume 38. Walter de Gruyter, 2011.
  33. T. Kolsrud. On a class of probabilistic integrodifferential equa- tions. In: Ideas and Methods in Mathematics and Physics. Vol. 1, Cambridge University Press, 1992, pp. 168-172.
  34. Y. G. Kondratiev and T. Kuna. Harmonic analysis on configu- ration spaces I. General theory. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5(2):201-233, 2002.
  35. Y. G. Kondratiev and O. Kutoviy. On the metrical properties of the configuration space. Math. Nachr., 279(7):774-783, 2006.
  36. J. L. López and C. Ferreira. Asymptotic expansions of general- ized Stieltjes transforms of algebraically decaying functions. Stud. Appl. Math., 108(2):187-215, 2002.
  37. F. Mainardi. Fractional Calculus and Waves in Linear Viscoelas- ticity: An Introduction to Mathematical Models. World Scientific, 2010.
  38. M. M. Meerschaert and H.-P. Scheffler. Stochastic model for ultraslow diffusion. Stochastic Process. Appl., 116(9):1215-1235, 2006.
  39. M. M. Meerschaert and H.-P. Scheffler. Triangular array limits for continuous time random walks. Stochastic Process. Appl. 118, 1606-1633, 2008.
  40. M. M. Meerschaert, H.-P. Scheffler, P.Kern. Limit theorems for continuous-time random walks with infinite mean waiting times. J. Appl. Probab. 41 (2004), 455-466.
  41. R. Metzler and J. Klafter. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep., 339(1):1- 77, November 2000.
  42. R. Metzler and J. Klafter. The restaurant at the end of the ran- dom walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A, 37(31):R161-R208, 2004.
  43. A. Mura, M.S. Taqqu and F. Mainardi. Non-Markovian diffusion equations and processes: Analysis and simulations. Physica A, 387: 5033-5064, 2008.
  44. J. Prüss. Evolutionary Integral Equations and Applications, vol- ume 87 of Monographs in Mathematics. Birkhäuser Verlag, Basel, 1993.
  45. E. Ya. Riekstyn'sh. Asymptotic Expansions of Integrals, volume 3. Zinatne, Riga, 1981. Russian.
  46. K.-I. Sato. Lévy Processes and Infinite Divisible Distributions. Cambridge University Press, Cambridge, 1999.
  47. E. Seneta. Regularly Varying Functions, volume 508 of Lect. Notes Math. Springer, 1976.
  48. D. Sornette. Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools. Springer Science & Business Media, 2006.
  49. R. L. Schilling, R. Song, and Z. Vondraček. Bernstein Functions: Theory and Applications. De Gruyter Studies in Mathematics. De Gruyter, Berlin, 2 edition, 2012.
  50. B. Toaldo. Convolution-type derivatives, hitting times of subor- dinators and time-changed C 0 -semigroups. Potential Anal. 42: 115-140, 2015.
  51. R. Wong. Asymptotic Approximations of Integrals, volume 34 of SIAM's Classics in Applied Mathematics. SIAM, 2001.