Fractional contact model in the continuum

Abstract

We consider the evolution of correlation functions in a non-Markov version of the contact model in the continuum. The memory effects are introduced by assuming the fractional evolution equation for the statistical dynamics. This leads to a behavior of time-dependent correlation functions, essentially different from the one known for the standard contact model.

Key words: Contact model in the continuum, correlation functions, Caputo-Djrbashian fractional derivative, Mittag-Leffler function.

Full Text

Article Information

Title	Fractional contact model in the continuum
Source	Methods Funct. Anal. Topology, Vol. 21 (2015), no. 2, 179–187
MathSciNet	3407909
zbMATH	06533475
Milestones	Recieved 04/09/2014; Revised 30/11/2014
Copyright	The Author(s) 2015 (CC BY-SA)

Authors Information

A. N. Kochubei
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

Yu. G. Kondratiev
Fakultat fur Mathematik, Universitat Bielefeld, Bielefeld, 33615, Germany

Citation Example

Anatoly N. Kochubei and Yuri G. Kondratiev, Fractional contact model in the continuum, Methods Funct. Anal. Topology 21 (2015), no. 2, 179–187.

BibTex

@article {MFAT753,
    AUTHOR = {Kochubei, Anatoly N. and Kondratiev, Yuri G.},
     TITLE = {Fractional contact model in the continuum},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {2},
     PAGES = {179–187},
      ISSN = {1029-3531},
  MRNUMBER = {3407909},
 ZBLNUMBER = {06533475},
       URL = {http://mfat.imath.kiev.ua/article/?id=753},
}

References

1. Emilia G. Bazhlekova, Subordination principle for fractional evolution equations, Fract. Calc. Appl. Anal. 3 (2000), no. 3, 213-230.  MathSciNet 
2. Mkhitar M. Djrbashian, Harmonic analysis and boundary value problems in the complex domain, Birkhauser Verlag, Basel, 1993.  MathSciNet  CrossRef
3. Dmitri Finkelshtein, Yuri Kondratiev, Oleksandr Kutoviy, Individual based model with competition in spatial ecology, SIAM J. Math. Anal. 41 (2009), no. 1, 297-317.  MathSciNet  CrossRef
4. Dmitri Finkelshtein, Yuri Kondratiev, Oleksandr Kutoviy, Semigroup approach to birth-and-death stochastic dynamics in continuum, J. Funct. Anal. 262 (2012), no. 3, 1274-1308.  MathSciNet  CrossRef
5. Dmitri L. Finkelshtein, Yuri G. Kondratiev, Maria Joao Oliveira, Markov evolutions and hierarchical equations in the continuum. I. One-component systems, J. Evol. Equ. 9 (2009), no. 2, 197-233.  MathSciNet  CrossRef
6. Rudolf Gorenflo, Yuri Luchko, Francesco Mainardi, Analytical properties and applications of the Wright function, Fract. Calc. Appl. Anal. 2 (1999), no. 4, 383-414.  MathSciNet 
7. Valentin Keyantuo, Carlos Lizama, Mahamadi Warma, Spectral criteria for solvability of boundary value problems and positivity of solutions of time-fractional differential equations, Abstr. Appl. Anal. (2013), Art. ID 614328, 11.  MathSciNet 
8. Anatoly A. Kilbas, Hari M. Srivastava, Juan J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam, 2006.  MathSciNet 
9. Yuri G. Kondratiev, Tobias Kuna, Harmonic analysis on configuration space. I. General theory, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5 (2002), no. 2, 201-233.  MathSciNet  CrossRef
10. Yuri Kondratiev, Oleksandr Kutoviy, Sergey Pirogov, Correlation functions and invariant measures in continuous contact model, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11 (2008), no. 2, 231-258.  MathSciNet  CrossRef
11. Yuri Kondratiev, Anatoli Skorokhod, On contact processes in continuum, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), no. 2, 187-198.  MathSciNet  CrossRef