With Andrzej Lasota there and back again
Abstract
The paper below is a written version of the 17th Andrzej Lasota Lecture presented on January 12th, 2024 in Katowice. During the lecture we tried to show the impact of Andrzej Lasota’s results on the author’s research concerning various fields of mathematics, including chaos and ergodicity of dynamical systems, Markov operators and semigroups and partial differentia equations.
Keywords
chaos; invariant measure; partial differential equation; Markov operator; semigroup of operators; asymptotic stability; piecewise deterministic Markov process; application to biological models
References
J. Auslander and J.A. Yorke, Interval maps, factors of maps and chaos, Tohoku Math. J. (2) 32 (1980), 177–188.
A. Bobrowski, T. Lipniacki, K. Pichór, and R. Rudnicki, Asymptotic behavior of distributions of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl. 333 (2007), 753–769.
A. Bobrowski and R. Rudnicki, On convergence and asymptotic behaviour of semigroups of operators, Philos. Trans. Roy. Soc. A 378 (2020), 20190613, 18 pp.
F. Comets, S. Popov, G.M. Schütz, and M. Vachkovskaia, Billiards in a general domain with random reflections, Arch. Ration. Mech. Anal. 191 (2009), 497–537.
M.H.A. Davis, Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models, J. Roy. Statist. Soc. Ser. B 46 (1984), 353–388.
R.L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley Stud. Nonlinearity, Addison-Wesley Publishing Company, Redwood City, CA, 1989.
H.M. Hilden and L.J. Wallen, Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math. J. 23 (1974), 557–565.
A. Lasota, Invariant measures and a linear model of turbulence, Rend. Sem. Mat. Univ. Padova 61 (1979), 40–48.
A. Lasota, Stable and chaotic solutions of a first order partial differential equation, Nonlinear Anal. 5 (1981), 1181–1193.
A. Lasota, Asymptotic stability of some nonlinear Boltzmann-type equations, J. Math. Anal. Appl. 268 (2002), 291–309.
A. Lasota and M.C. Mackey, Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, Appl. Math. Sci., 97, Springer-Verlag, New York, 1994.
A. Lasota and R. Rudnicki, Asymptotic behaviour of semigroups of positive operators on C(X), Bull. Polish Acad. Sci. Math. 36 (1988), 151–159.
A. Lasota and J. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481–488.
A. Lasota and J. Yorke, On the existence of invariant measures for transformations with strictly turbulent trajectories, Bull. Polish Acad. Sci. Math. 25 (1977), 233–238.
A. Lasota and J. Yorke, Exact dynamical systems and the Frobenius–Perron operator, Trans. Amer. Math. Soc. 273 (1982), 375–384.
A. Lasota and J. Yorke, When the long time behavior is independent of the initial density, SIAM J. Math. Anal. 27 (1996), 221–240.
B. Lods, M. Mokhtar-Kharroubi, and R. Rudnicki, Invariant density and time asymptotics for collisionless kinetic equations with partly diffuse boundary operators, Ann. Inst. H. Poincaré C Anal. Non Linéaire 37 (2020), 877–923.
M.C. Mackey and R. Rudnicki, Asymptotic similarity and Malthusian growth in autonomous and nonautonomous populations, J. Math. Anal. Appl. 187 (1994), 548–566.
M.C. Mackey and R. Rudnicki, A new criterion for global stability of cell simultaneous cell replication and maturation processes, J. Math. Biol. 38 (1999), 195–219.
M. Mokhtar-Kharroubi and R. Rudnicki, On asymptotic stability and sweeping of collisionless kinetic equations, Acta Appl. Math. 147 (2017), 19–38.
G. Pianigiani and J.A. Yorke, Expanding maps on sets which are almost invariant: decay and chaos, Trans. Amer. Math. Soc. 252 (1979), 351–366.
K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl. 249 (2000), 668–685.
K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic operators and semigroups, J. Math. Anal. Appl. 436 (2016), 305–321.
K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic semigroups and applications, Stoch. Dyn. 18 (2018), 1850001, 18 pp.
K. Pichór and R. Rudnicki, Stability of stochastic semigroups and applications to Stein’s neuronal model, Discrete Contin. Dyn. Syst. Ser. B 23 (2018), 377–385.
K. Pichór and R. Rudnicki, Applications of stochastic semigroups to cell cycle models, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), 2365–2381.
K. Pichór and R. Rudnicki, Dynamics of antibody levels: asymptotic properties, Math. Methods Appl. Sci. 43 (2020), 10490–10499.
K. Pichór and R. Rudnicki, Cell cycle length and long-time behavior of an age-size model, Math. Methods Appl. Sci. 45 (2022), 5797–5820.
K. Pichór and R. Rudnicki, Asymptotic properties of a general model of immune status, SIAM J. Appl. Math. 83 (2023), 172–193.
R. Rudnicki, Invariant measures for the flow of a first order partial differential equation, Ergodic Theory Dynam. Systems 5 (1985), 437–443.
R. Rudnicki, Asymptotic properties of the iterates of positive operators on C(X), Bull. Polish Acad. Sci. Math. 34 (1986), 181–187.
R. Rudnicki, Strong ergodic properties of a first-order partial differential equation, J. Math. Anal. Appl. 133 (1988), 14–26.
R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math. 43 (1995), 245–262.
R. Rudnicki, Chaos for some infinite-dimensional dynamical systems, Math. Methods Appl. Sci. 27 (2004), 723–738.
R. Rudnicki, Chaoticity of the blood cell production system, Chaos 19 (2009), 043112, 6 pp.
R. Rudnicki, Chaoticity and invariant measures for a cell population model, J. Math. Anal. Appl. 393 (2012), 151–165.
R. Rudnicki, An ergodic theory approach to chaos, Discrete Contin. Dyn. Syst. 35 (2015), 757–770.
R. Rudnicki, Models and Methods Mathematical Biology. Part II: Probabilistic Models, (in Polish), Księgozbiór Matematyczny 4, IMPAN, Warszawa, 2022. (Modele i Metody Biologii Matematycznej. Część II: Modele Probabilistyczne.)
R. Rudnicki, Ergodic properties of a semilinear partial differential equation, J. Differential Equations 372 (2023), 235–253.
R. Rudnicki and A. Tomski, On a stochastic gene expression with pre-mRNA, mRNA and protein contribution, J. Theoret. Biol. 387 (2015), 54–67.
R. Rudnicki and M. Tyran-Kamińska, Piecewise Deterministic Processes in Biological Models, SpringerBriefs Appl. Sci. Technol., Math. Methods, Springer, Cham, 2017.
R. Rudnicki and P. Zwoleński, Model of phenotypic evolution in hermaphroditic populations, J. Math. Biol. 70 (2015), 1295–1321.
Instytut Matematyczny Polskiej Akademii Nauk, Katowice Poland
https://orcid.org/0000-0002-3042-5662
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