# American Institute of Mathematical Sciences

September  2017, 10(3): 725-740. doi: 10.3934/krm.2017029

## Fractional kinetic hierarchies and intermittency

 1 Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska 3, Kyiv, 01004, Ukraine 2 Department of Mathematics, University of Bielefeld, D-33615 Bielefeld, Germany

Received  June 2016 Revised  October 2016 Published  December 2016

We consider general convolutional derivatives and related fractional statistical dynamics of continuous interacting particle systems. We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. Conditions for the intermittency property of fractional kinetic dynamics are obtained.

Citation: Anatoly N. Kochubei, Yuri Kondratiev. Fractional kinetic hierarchies and intermittency. Kinetic and Related Models, 2017, 10 (3) : 725-740. doi: 10.3934/krm.2017029
##### References:
 [1] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2011. doi: 10.1007/978-3-0348-0087-7. [2] C. Batty, R. Chill and Yu. Tomilov, Fine scale of decay of operator semigroups, J. European Math. Soc., 18 (2016), 853-929.  doi: 10.4171/JEMS/605. [3] E. Bazhlekova, Subordination principle for fractional evolution equations, Frac. Calc. Appl. Anal., 3 (2000), 213-230. [4] E. Bazhlekova, Fractional Evolution Equations in Banach Spaces. Ph. D. Thesis, Eindhoven University of Technology, 2001. [5] E. Bazhlekova, Completely monotone functions and some classes of fractional evolution equations, Integr. Transf. and Special Funct., 26 (2015), 737-752.  doi: 10.1080/10652469.2015.1039224. [6] N. N. Bogoliubov, Problems of a dynamical theory in statistical physics, (Russian), Gostekhisdat, Moscow, 1946. English translation in Studies in Statistical Mechanics (J. de Boer and G. E. Uhlenbeck, eds), volume 1, pages 1-118, North-Holland, Amsterdam, 1962. [7] R. Carmona and S. A. Molchanov, Parabolic Anderson Problem and Intermittency, Memoirs of the American Mathematical Society, Vol. 518, American Mathematical Soc., 1994. doi: 10.1090/memo/0518. [8] R. A. Carmona and S. A. Molchanov, Stationary parabolic Anderson model and intermittency, Probab. Theory Related Fields, 102 (1995), 433-453.  doi: 10.1007/BF01198845. [9] A. V. Chechkin, R. Gorenflo, I. M. Sokolov and V. Yu. Gonchar, Distributed order fractional diffusion equation, Fract. Calc. Appl. Anal., 6 (2003), 259-279. [10] J. L. Da Silva, A. N. Kochubei and Y. Kondratiev, Fractional statistical dynamics and kinetic equations, Methods Funct. Anal. Topology, 22 (2016), 197-209. [11] M. M. Djrbashian, Integral Transformations and Representations of Functions on a Complex Domain, Nauka, Moscow, 1966 (Russian). [12] G. Doetsch, Introduction to the Theory and Applications of the Laplace Transformation, Springer, Berlin, 1974. [13] S. D. Eidelman, S. D. Ivasyshen and A. N. Kochubei. Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Birkhäuser, Basel, 2004. doi: 10.1007/978-3-0348-7844-9. [14] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York, 1971. [15] D. L. Finkelshtein, Y. G. Kondratiev and O. Kutoviy, Vlasov scaling for stochastic dynamics of continuous systems, J. Stat. Phys., 141 (2010), 158-178.  doi: 10.1007/s10955-010-0038-1. [16] D. Finkelshtein, Y. G. Kondratiev and O. Kutoviy, Semigroup approach to birth-and-death stochastic dynamics in continuum, J. Funct. Anal., 262 (2012), 1274-1308.  doi: 10.1016/j.jfa.2011.11.005. [17] D. Finkelshtein, Y. G. Kondratiev and O. Kutoviy, Statistical dynamics of continuous systems: Perturbative and approximative approaches, Arab. J. Math., 4 (2015), 255-300.  doi: 10.1007/s40065-014-0111-8. [18] D. Finkelshtein, Y. G. Kondratiev, Y. Kozitsky and O. Kutoviy, The statistical dynamics of a spatial logistic model and the related kinetic equation, Math. Models Methods Appl. Sci., 25 (2015), 343-370.  doi: 10.1142/S0218202515500128. [19] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, 2014. doi: 10.1007/978-3-662-43930-2. [20] G. Gripenberg, Volterra integro-differential equations with accretive nonlinearity, J. Diff. Equat., 60 (1985), 57-79.  doi: 10.1016/0022-0396(85)90120-2. [21] N. Jacob, Pseudo-Differential Operators and Markov Processes, Vol. 1, London, Imperial College Press, 2001.  doi: 10.1142/9781860949746. [22] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [23] A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024. [24] A. N. Kochubei, Distributed order derivatives and relaxation patterns, J. Phys. A, 42 (2009), Article 315203, 9pp.  doi: 10.1088/1751-8113/42/31/315203. [25] A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integral Equations Oper. Theory, 71 (2011), 583-600.  doi: 10.1007/s00020-011-1918-8. [26] V. N. Kolokoltsov, Markov Processes, Semigroups and Generators, De Gruyter, Berlin, 2011. [27] Y. G. Kondratiev and T. Kuna, Harmonic analysis on configuration spaces. Ⅰ. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5 (2002), 201-233.  doi: 10.1142/S0219025702000833. [28] Y. G. Kondratiev and O. Kutoviy, On the metrical properties of the configuration space, Math. Nachr., 279 (2006), 774-783.  doi: 10.1002/mana.200310392. [29] Y. G. Kondratiev, O. Kutoviy and S. Pirogov, Correlation functions and invariant measures in continuous contact model, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 11 (2008), 231-258.  doi: 10.1142/S0219025708003038. [30] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.  doi: 10.1142/9781848163300. [31] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3. [32] R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), R161-R208.  doi: 10.1088/0305-4470/37/31/R01. [33] L. P. ∅sterdal, Subadditive functions and their (pseudo)-inverses, J. Math. Anal. Appl., 317 (2006), 724-731.  doi: 10.1016/j.jmaa.2005.05.039. [34] J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. [35] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999. [36] R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, Walter de Gruyter, Berlin, 2010. [37] E. Seneta, Regularly Varying Functions, Lecture Notes Math. 508, 1976. [38] H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615.  doi: 10.1103/RevModPhys.52.569.

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##### References:
 [1] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2011. doi: 10.1007/978-3-0348-0087-7. [2] C. Batty, R. Chill and Yu. Tomilov, Fine scale of decay of operator semigroups, J. European Math. Soc., 18 (2016), 853-929.  doi: 10.4171/JEMS/605. [3] E. Bazhlekova, Subordination principle for fractional evolution equations, Frac. Calc. Appl. Anal., 3 (2000), 213-230. [4] E. Bazhlekova, Fractional Evolution Equations in Banach Spaces. Ph. D. Thesis, Eindhoven University of Technology, 2001. [5] E. Bazhlekova, Completely monotone functions and some classes of fractional evolution equations, Integr. Transf. and Special Funct., 26 (2015), 737-752.  doi: 10.1080/10652469.2015.1039224. [6] N. N. Bogoliubov, Problems of a dynamical theory in statistical physics, (Russian), Gostekhisdat, Moscow, 1946. English translation in Studies in Statistical Mechanics (J. de Boer and G. E. Uhlenbeck, eds), volume 1, pages 1-118, North-Holland, Amsterdam, 1962. [7] R. Carmona and S. A. Molchanov, Parabolic Anderson Problem and Intermittency, Memoirs of the American Mathematical Society, Vol. 518, American Mathematical Soc., 1994. doi: 10.1090/memo/0518. [8] R. A. Carmona and S. A. Molchanov, Stationary parabolic Anderson model and intermittency, Probab. Theory Related Fields, 102 (1995), 433-453.  doi: 10.1007/BF01198845. [9] A. V. Chechkin, R. Gorenflo, I. M. Sokolov and V. Yu. Gonchar, Distributed order fractional diffusion equation, Fract. Calc. Appl. Anal., 6 (2003), 259-279. [10] J. L. Da Silva, A. N. Kochubei and Y. Kondratiev, Fractional statistical dynamics and kinetic equations, Methods Funct. Anal. Topology, 22 (2016), 197-209. [11] M. M. Djrbashian, Integral Transformations and Representations of Functions on a Complex Domain, Nauka, Moscow, 1966 (Russian). [12] G. Doetsch, Introduction to the Theory and Applications of the Laplace Transformation, Springer, Berlin, 1974. [13] S. D. Eidelman, S. D. Ivasyshen and A. N. Kochubei. Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Birkhäuser, Basel, 2004. doi: 10.1007/978-3-0348-7844-9. [14] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York, 1971. [15] D. L. Finkelshtein, Y. G. Kondratiev and O. Kutoviy, Vlasov scaling for stochastic dynamics of continuous systems, J. Stat. Phys., 141 (2010), 158-178.  doi: 10.1007/s10955-010-0038-1. [16] D. Finkelshtein, Y. G. Kondratiev and O. Kutoviy, Semigroup approach to birth-and-death stochastic dynamics in continuum, J. Funct. Anal., 262 (2012), 1274-1308.  doi: 10.1016/j.jfa.2011.11.005. [17] D. Finkelshtein, Y. G. Kondratiev and O. Kutoviy, Statistical dynamics of continuous systems: Perturbative and approximative approaches, Arab. J. Math., 4 (2015), 255-300.  doi: 10.1007/s40065-014-0111-8. [18] D. Finkelshtein, Y. G. Kondratiev, Y. Kozitsky and O. Kutoviy, The statistical dynamics of a spatial logistic model and the related kinetic equation, Math. Models Methods Appl. Sci., 25 (2015), 343-370.  doi: 10.1142/S0218202515500128. [19] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, 2014. doi: 10.1007/978-3-662-43930-2. [20] G. Gripenberg, Volterra integro-differential equations with accretive nonlinearity, J. Diff. Equat., 60 (1985), 57-79.  doi: 10.1016/0022-0396(85)90120-2. [21] N. Jacob, Pseudo-Differential Operators and Markov Processes, Vol. 1, London, Imperial College Press, 2001.  doi: 10.1142/9781860949746. [22] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [23] A. N. Kochubei, Distributed order calculus and equations of ultraslow diffusion, J. Math. Anal. Appl., 340 (2008), 252-281.  doi: 10.1016/j.jmaa.2007.08.024. [24] A. N. Kochubei, Distributed order derivatives and relaxation patterns, J. Phys. A, 42 (2009), Article 315203, 9pp.  doi: 10.1088/1751-8113/42/31/315203. [25] A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integral Equations Oper. Theory, 71 (2011), 583-600.  doi: 10.1007/s00020-011-1918-8. [26] V. N. Kolokoltsov, Markov Processes, Semigroups and Generators, De Gruyter, Berlin, 2011. [27] Y. G. Kondratiev and T. Kuna, Harmonic analysis on configuration spaces. Ⅰ. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 5 (2002), 201-233.  doi: 10.1142/S0219025702000833. [28] Y. G. Kondratiev and O. Kutoviy, On the metrical properties of the configuration space, Math. Nachr., 279 (2006), 774-783.  doi: 10.1002/mana.200310392. [29] Y. G. Kondratiev, O. Kutoviy and S. Pirogov, Correlation functions and invariant measures in continuous contact model, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 11 (2008), 231-258.  doi: 10.1142/S0219025708003038. [30] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.  doi: 10.1142/9781848163300. [31] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Physics Reports, 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3. [32] R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), R161-R208.  doi: 10.1088/0305-4470/37/31/R01. [33] L. P. ∅sterdal, Subadditive functions and their (pseudo)-inverses, J. Math. Anal. Appl., 317 (2006), 724-731.  doi: 10.1016/j.jmaa.2005.05.039. [34] J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. [35] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999. [36] R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, Walter de Gruyter, Berlin, 2010. [37] E. Seneta, Regularly Varying Functions, Lecture Notes Math. 508, 1976. [38] H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615.  doi: 10.1103/RevModPhys.52.569.
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