Markov operators: applications to diffusion processes and population dynamics
Applicationes Mathematicae (2000)
- Volume: 27, Issue: 1, page 67-79
- ISSN: 1233-7234
Access Full Article
topAbstract
topHow to cite
topRudnicki, Ryszard. "Markov operators: applications to diffusion processes and population dynamics." Applicationes Mathematicae 27.1 (2000): 67-79. <http://eudml.org/doc/219260>.
@article{Rudnicki2000,
abstract = {This note contains a survey of recent results concerning asymptotic properties of Markov operators and semigroups. Some biological and physical applications are given.},
author = {Rudnicki, Ryszard},
journal = {Applicationes Mathematicae},
keywords = {Markov operator; asymptotic stability; diffusion process; partial differential equation; asymptotic properties; Markov operators and semigroups; sweeping; Foguel alternative; Khasminskij function; differential equations; diffusion and jump processes},
language = {eng},
number = {1},
pages = {67-79},
title = {Markov operators: applications to diffusion processes and population dynamics},
url = {http://eudml.org/doc/219260},
volume = {27},
year = {2000},
}
TY - JOUR
AU - Rudnicki, Ryszard
TI - Markov operators: applications to diffusion processes and population dynamics
JO - Applicationes Mathematicae
PY - 2000
VL - 27
IS - 1
SP - 67
EP - 79
AB - This note contains a survey of recent results concerning asymptotic properties of Markov operators and semigroups. Some biological and physical applications are given.
LA - eng
KW - Markov operator; asymptotic stability; diffusion process; partial differential equation; asymptotic properties; Markov operators and semigroups; sweeping; Foguel alternative; Khasminskij function; differential equations; diffusion and jump processes
UR - http://eudml.org/doc/219260
ER -
References
top- [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. Zbl0839.47021
- [3] W. Bartoszek and T. Brown, On Frobenius-Perron operators which overlap supports, Bull. Polish Acad. Sci. Math. 45 (1997), 17-24. Zbl0891.47006
- [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. Zbl0770.60094
- [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. Zbl0543.92021
- [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. Zbl0767.47012
- [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. Zbl0529.92011
- [12] J. Łuczka and R. Rudnicki, Randomly flashing diffusion: asymptotic properties, J. Statist. Phys. 83 (1996), 1149-1164. Zbl1081.82602
- [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. Zbl0833.92014
- [14] J. Malczak, An application of Markov operators in differential and integral equations, Rend. Sem. Mat. Univ. Padova 87 (1992), 281-297. Zbl0795.60071
- [15] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomath. 68, Springer, New York, 1986. Zbl0614.92014
- [16] K. Pichór, Asymptotic stability of a partial differential equation with an integral perturbation, Ann. Polon. Math. 68 (1998), 83-96, Zbl0911.45005
- [17] K. Pichór and R. Rudnicki, Stability of Markov semigroups and applications to parabolic systems, J. Math. Anal. Appl. 215 (1997), 56-74. Zbl0892.35072
- [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. Zbl0909.47032
- [19] R. Rudnicki, Asymptotic behaviour of a transport equation, Ann. Polon. Math. 57 (1992), 45-55. Zbl0758.45009
- [20] R. Rudnicki, Asymptotic behaviour of an integro-parabolic equation, Bull. Polish Acad. Sci. Math. 40 (1992), 111-128. Zbl0767.45007
- [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. Zbl0839.35013
- [22] R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math. 43 (1995), 245-262. Zbl0838.47040
- [23] R. Rudnicki, Asymptotic stability of Markov operators: a counter-example, ibid. 45 (1997), 1-5. Zbl0891.47007
- [24] R. Rudnicki and K. Pichór, Markov semigroups and stability of the cell maturity distribution, J. Biol. Systems, submitted. Zbl0965.47026
- [25] J. Traple, Markov semigroups generated by Poisson driven differential equations, Bull. Polish Acad. Sci. Math. 44 (1996), 230-252. Zbl0861.45008
- [26] J. Tyrcha, Asymptotic stability in a generalized probabilistic/deterministic model of the cell cycle, J. Math. Biol. 26 (1988), 465-475. Zbl0716.92017
- [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. Zbl0582.92020
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.