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 & 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. Google Scholar

[2]

C. BattyR. Chill and Yu. Tomilov, Fine scale of decay of operator semigroups, J. European Math. Soc., 18 (2016), 853-929. doi: 10.4171/JEMS/605. Google Scholar

[3]

E. Bazhlekova, Subordination principle for fractional evolution equations, Frac. Calc. Appl. Anal., 3 (2000), 213-230. Google Scholar

[4]

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces. Ph. D. Thesis, Eindhoven University of Technology, 2001.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

A. V. ChechkinR. GorenfloI. M. Sokolov and V. Yu. Gonchar, Distributed order fractional diffusion equation, Fract. Calc. Appl. Anal., 6 (2003), 259-279. Google Scholar

[10]

J. L. Da SilvaA. N. Kochubei and Y. Kondratiev, Fractional statistical dynamics and kinetic equations, Methods Funct. Anal. Topology, 22 (2016), 197-209. Google Scholar

[11]

M. M. Djrbashian, Integral Transformations and Representations of Functions on a Complex Domain, Nauka, Moscow, 1966 (Russian). Google Scholar

[12]

G. Doetsch, Introduction to the Theory and Applications of the Laplace Transformation, Springer, Berlin, 1974. Google Scholar

[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. Google Scholar

[14]

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York, 1971. Google Scholar

[15]

D. L. FinkelshteinY. 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. Google Scholar

[16]

D. FinkelshteinY. 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. Google Scholar

[17]

D. FinkelshteinY. 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. Google Scholar

[18]

D. FinkelshteinY. G. KondratievY. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21] N. Jacob, Pseudo-Differential Operators and Markov Processes, Vol. 1, London, Imperial College Press, 2001. doi: 10.1142/9781860949746. Google Scholar
[22]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

V. N. Kolokoltsov, Markov Processes, Semigroups and Generators, De Gruyter, Berlin, 2011. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

Y. G. KondratievO. 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. Google Scholar

[30] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. doi: 10.1142/9781848163300. Google Scholar
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. Google Scholar

[35] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999. Google Scholar
[36]

R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, Walter de Gruyter, Berlin, 2010. Google Scholar

[37]

E. Seneta, Regularly Varying Functions, Lecture Notes Math. 508, 1976. Google Scholar

[38]

H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615. doi: 10.1103/RevModPhys.52.569. Google Scholar

show all references

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. Google Scholar

[2]

C. BattyR. Chill and Yu. Tomilov, Fine scale of decay of operator semigroups, J. European Math. Soc., 18 (2016), 853-929. doi: 10.4171/JEMS/605. Google Scholar

[3]

E. Bazhlekova, Subordination principle for fractional evolution equations, Frac. Calc. Appl. Anal., 3 (2000), 213-230. Google Scholar

[4]

E. Bazhlekova, Fractional Evolution Equations in Banach Spaces. Ph. D. Thesis, Eindhoven University of Technology, 2001.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

A. V. ChechkinR. GorenfloI. M. Sokolov and V. Yu. Gonchar, Distributed order fractional diffusion equation, Fract. Calc. Appl. Anal., 6 (2003), 259-279. Google Scholar

[10]

J. L. Da SilvaA. N. Kochubei and Y. Kondratiev, Fractional statistical dynamics and kinetic equations, Methods Funct. Anal. Topology, 22 (2016), 197-209. Google Scholar

[11]

M. M. Djrbashian, Integral Transformations and Representations of Functions on a Complex Domain, Nauka, Moscow, 1966 (Russian). Google Scholar

[12]

G. Doetsch, Introduction to the Theory and Applications of the Laplace Transformation, Springer, Berlin, 1974. Google Scholar

[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. Google Scholar

[14]

W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, Wiley, New York, 1971. Google Scholar

[15]

D. L. FinkelshteinY. 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. Google Scholar

[16]

D. FinkelshteinY. 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. Google Scholar

[17]

D. FinkelshteinY. 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. Google Scholar

[18]

D. FinkelshteinY. G. KondratievY. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21] N. Jacob, Pseudo-Differential Operators and Markov Processes, Vol. 1, London, Imperial College Press, 2001. doi: 10.1142/9781860949746. Google Scholar
[22]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

V. N. Kolokoltsov, Markov Processes, Semigroups and Generators, De Gruyter, Berlin, 2011. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

Y. G. KondratievO. 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. Google Scholar

[30] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010. doi: 10.1142/9781848163300. Google Scholar
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[34]

J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. Google Scholar

[35] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999. Google Scholar
[36]

R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, Walter de Gruyter, Berlin, 2010. Google Scholar

[37]

E. Seneta, Regularly Varying Functions, Lecture Notes Math. 508, 1976. Google Scholar

[38]

H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys., 52 (1980), 569-615. doi: 10.1103/RevModPhys.52.569. Google Scholar

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