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2000 | 27 | 1 | 67-79
Tytuł artykułu

Markov operators: applications to diffusion processes and population dynamics

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
This note contains a survey of recent results concerning asymptotic properties of Markov operators and semigroups. Some biological and physical applications are given.
Rocznik
Tom
27
Numer
1
Strony
67-79
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-01-15
Twórcy
  • Institute of Mathematics, Polish Academy of Sciences, Staromiejska 8/6 40-013 Katowice, Poland
Bibliografia
  • [1] V. Balakrishnan, C. Van den Broeck and P. Hanggi, First-passage times of non-Markovian processes: the case of a reflecting boundary, Phys. Rev. A 38 (1988), 4213-4222.
  • [2] K. Baron and A. Lasota, Asymptotic properties of Markov operators defined by Volterra type integrals, Ann. Polon. Math. 58 (1993), 161-175.
  • [3] W. Bartoszek and T. Brown, On Frobenius-Perron operators which overlap supports, Bull. Polish Acad. Sci. Math. 45 (1997), 17-24.
  • [4] V. Bezak, A modification of the Wiener process due to a Poisson random train of diffusion-enhancing pulses, J. Phys. A 25 (1992), 6027-6041.
  • [5] S. Chandrasekhar and G. Münch, The theory of fluctuations in brightness of the Milky-Way, Astrophys. J. 125, 94-123.
  • [6] O. Diekmann, H. J. A. Heijmans and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol. 19 (1984), 227-248.
  • [7] S. R. Foguel, The Ergodic Theory of Markov Processes, Van Nostrand Reinhold, New York, 1969.
  • [8] R. Z. Hasminskiĭ, Ergodic properties of recurrent diffusion processes and stabilization of the solutions of the Cauchy problem for parabolic equations, Teor. Veroyatnost. Primen. 5 (1960), 196-214 (in Russian).
  • [9] T. Komorowski and J. Tyrcha, Asymptotic properties of some Markov operators, Bull. Polish Acad. Sci. Math. 37 (1989), 221-228.
  • [10] A. Lasota and M. C. Mackey, Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, Appl. Math. Sci. 97, Springer, New York, 1994.
  • [11] A. Lasota and M. C. Mackey, Globally asymptotic properties of proliferating cell populations, J. Math. Biol. 19 (1984), 43-62.
  • [12] J. Łuczka and R. Rudnicki, Randomly flashing diffusion: asymptotic properties, J. Statist. Phys. 83 (1996), 1149-1164.
  • [13] M. C. Mackey and R. Rudnicki, Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol. 33 (1994), 89-109.
  • [14] J. Malczak, An application of Markov operators in differential and integral equations, Rend. Sem. Mat. Univ. Padova 87 (1992), 281-297.
  • [15] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomath. 68, Springer, New York, 1986.
  • [16] K. Pichór, Asymptotic stability of a partial differential equation with an integral perturbation, Ann. Polon. Math. 68 (1998), 83-96,
  • [17] K. Pichór and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems, J. Math. Anal. Appl. 215 (1997), 56-74.
  • [18] K. Pichór and R. Rudnicki, Asymptotic behaviour of Markov semigroups and applications to transport equations, Bull. Polish Acad. Sci. Math. 45 (1997), 379-397.
  • [19] R. Rudnicki, Asymptotic behaviour of a transport equation, Ann. Polon. Math. 57 (1992), 45-55.
  • [20] R. Rudnicki, Asymptotic behaviour of an integro-parabolic equation, Bull. Polish Acad. Sci. Math. 40 (1992), 111-128.
  • [21] R. Rudnicki, Asymptotic properties of the Fokker-Planck equation, in: Chaos--The Interplay between Stochastics and Deterministic Behaviour, Karpacz '95 Proc., P. Garbaczewski, M. Wolf and A. Weron (eds.), Lecture Notes in Phys. 457, Springer, Berlin, 1995, 517-521.
  • [22] R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math. 43 (1995), 245-262.
  • [23] R. Rudnicki, Asymptotic stability of Markov operators: a counter-example, ibid. 45 (1997), 1-5.
  • [24] R. Rudnicki and K. Pichór, Markov semigroups and stability of the cell maturity distribution, J. Biol. Systems, submitted.
  • [25] J. Traple, Markov semigroups generated by Poisson driven differential equations, Bull. Polish Acad. Sci. Math. 44 (1996), 230-252.
  • [26] J. Tyrcha, Asymptotic stability in a generalized probabilistic/deterministic model of the cell cycle, J. Math. Biol. 26 (1988), 465-475.
  • [27] J. J. Tyson and K. B. Hannsgen, Cell growth and division: A deterministic/probabilistic model of the cell cycle, ibid. 23 (1986), 231-246.
Typ dokumentu
Bibliografia
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