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On the Separating Variables Method for Markov Death-Process Equations

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Abstract

We consider a method of obtaining non-closed solutions of the first and second Kolmogorov equations for the exponential (double) generating function of transition probabilities for quadratic death-processes of one, two and three dimensions. We obtain a representation for the generating function of transition probabilities in the form of a Fourier series, using generalized hypergeometric functions and Jacobi polynomials.

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References

  1. Anderson, W.J.: Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer, New York (1991)

    Book  MATH  Google Scholar 

  2. Athreya, K.B., Ney, P.: Branching Processes. Springer, New York (1972)

    Book  MATH  Google Scholar 

  3. Becker, N.G.: Interactions between species: some comparisons between deterministic and stochastic models. Rocky Mt. J. Math. 3, 53–68 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, A., Li, J., Chen, Y., Zhou, D.: Extinction probability of interacting branching collision processes. Adv. Appl. Prob. 44, 226–259 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dadvey, I.G., Ninham, B.W., Staff, P.J.: Stochastic models for second-order chemical reaction kinetics. Equilib. State. J. Chem. Phys. 45, 2145–2155 (1966)

    Google Scholar 

  6. Daley, D.J., Gani, J.: A random allocation model for carrier-borne epidemics. J. Appl. Prob. 30, 751–765 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Erdelyi, A. (ed.): Higher Transcendental Functions, vol. 2. McGraw-Hill, New York (1953)

    MATH  Google Scholar 

  8. Erdelyi, A. (ed.): Higher Transcendental Functions, vol. 3. McGraw-Hill, New York (1955)

    MATH  Google Scholar 

  9. Gel’fond, A.O., Leont’ev, A.F.: A generalization of Fourier series. Mat. Sb. 29, 477–500 (1951). (in Russian)

    MathSciNet  Google Scholar 

  10. Gikhman, I.I., Skorokhod, A.V.: Introduction to the Theory of Random Processes, 2nd edn. Dover Publ., New York (1996)

    MATH  Google Scholar 

  11. Johnson, N.L., Kotz, S. (eds.) Two-sex problem. In: Encyclopaedia of Statistical Sciences. vol. 9. Wiley, New York (1988)

  12. Kalinkin, A.V.: Markov branching processes with interaction. Russ. Math. Surv. 57, 241–304 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kalinkin, A.V., Mastikhin, A.V.: A limit theorem for a Weiss epidemic process. J. Appl. Prob. 52, 247–257 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kamke, E.: Differentialgleichungen, lösungsmethoden und lösungen, 4th edn. Geest und Portig, Leipzig (1959)

    MATH  Google Scholar 

  15. Lederman, W., Reuter, G.E.H.: Spectral theory for the differential equations of simple birth and death processes. Philos. Trans. R. Soc. Lond. Ser. A 246, 321–369 (1954)

    Article  MathSciNet  Google Scholar 

  16. Letessier, J., Valent, G.: Exact eigenfunctions and spectrum for several cubic and quartic birth and death processes. Phys. Lett. Ser. A 108, 245–247 (1985)

    Article  MathSciNet  Google Scholar 

  17. Letessier, J., Valent, G.: Some exact solutions of the Kolmogorov boundary value problem. Approx. Theory Appl. 4, 97–117 (1988)

    MathSciNet  MATH  Google Scholar 

  18. Mastikhin, A.V.: Final distribution for Gani epidemic Markov process. Math. Notes 82, 787–797 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. McQuarrie, D.A., Jachimowcki, C.J., Russel, M.E.: Kinetic of small system. II. J. Chem. Phys. 40, 2914–2921 (1964)

    Article  Google Scholar 

  20. McQuarrie, D.A.: Stochastic approach to chemical kinetics. J. Appl. Prob. 4, 413–478 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  21. Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series. More Special Functions, vol. 3. Gordon and Breach, New York (1989)

    MATH  Google Scholar 

  22. Sevast’yanov, B.A.: Vetvyaščiesya protsessy (Branching processes). Nauka, Moscow (1971) (in Russian) (Also available in German: Sewastjanow, B.A.: Verzweigungsprozesse. Akademie-Verlag, Berlin (1974))

  23. Turkina L.V.: Solution of the Kolmogorov equations for Markov birth-processes of quadratic type. Graduation thesis, Bauman Moscow State Technical University, p 99 (2008) (in Russian)

  24. Valent, G.: An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17, 688–703 (1986)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of ‘Higher Mathematics’, Bauman Moscow State Technical University, 2-nd Baumanskaya Street, 5, Moscow, Russia, 105005

    Aleksandr V. Kalinkin & Anton V. Mastikhin

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  1. Aleksandr V. Kalinkin
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  2. Anton V. Mastikhin
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Correspondence to Aleksandr V. Kalinkin.

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Kalinkin, A.V., Mastikhin, A.V. On the Separating Variables Method for Markov Death-Process Equations. J Theor Probab 32, 163–182 (2019). https://doi.org/10.1007/s10959-017-0795-8

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  • DOI: https://doi.org/10.1007/s10959-017-0795-8

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