With Andrzej Lasota there and back again

Ryszard Rudnicki

Published: 2024-07-15

Vol. 38 No. 2 (2024)

With Andrzej Lasota there and back again

Ryszard Rudnicki

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

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Rudnicki, R. (2024). With Andrzej Lasota there and back again. Annales Mathematicae Silesianae, 38(2), 134–154. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/17705

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References

J. Auslander and J.A. Yorke, Interval maps, factors of maps and chaos, Tohoku Math. J. (2) 32 (1980), 177–188.
Google Scholar

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.
Google Scholar

A. Bobrowski and R. Rudnicki, On convergence and asymptotic behaviour of semigroups of operators, Philos. Trans. Roy. Soc. A 378 (2020), 20190613, 18 pp.
Google Scholar

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.
Google Scholar

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.
Google Scholar

R.L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley Stud. Nonlinearity, Addison-Wesley Publishing Company, Redwood City, CA, 1989.
Google Scholar

H.M. Hilden and L.J. Wallen, Some cyclic and non-cyclic vectors of certain operators, Indiana Univ. Math. J. 23 (1974), 557–565.
Google Scholar

A. Lasota, Invariant measures and a linear model of turbulence, Rend. Sem. Mat. Univ. Padova 61 (1979), 40–48.
Google Scholar

A. Lasota, Stable and chaotic solutions of a first order partial differential equation, Nonlinear Anal. 5 (1981), 1181–1193.
Google Scholar

A. Lasota, Asymptotic stability of some nonlinear Boltzmann-type equations, J. Math. Anal. Appl. 268 (2002), 291–309.
Google Scholar

A. Lasota and M.C. Mackey, Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, Appl. Math. Sci., 97, Springer-Verlag, New York, 1994.
Google Scholar

A. Lasota and R. Rudnicki, Asymptotic behaviour of semigroups of positive operators on C(X), Bull. Polish Acad. Sci. Math. 36 (1988), 151–159.
Google Scholar

A. Lasota and J. Yorke, On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc. 186 (1973), 481–488.
Google Scholar

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.
Google Scholar

A. Lasota and J. Yorke, Exact dynamical systems and the Frobenius–Perron operator, Trans. Amer. Math. Soc. 273 (1982), 375–384.
Google Scholar

A. Lasota and J. Yorke, When the long time behavior is independent of the initial density, SIAM J. Math. Anal. 27 (1996), 221–240.
Google Scholar

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.
Google Scholar

M.C. Mackey and R. Rudnicki, Asymptotic similarity and Malthusian growth in autonomous and nonautonomous populations, J. Math. Anal. Appl. 187 (1994), 548–566.
Google Scholar

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.
Google Scholar

M. Mokhtar-Kharroubi and R. Rudnicki, On asymptotic stability and sweeping of collisionless kinetic equations, Acta Appl. Math. 147 (2017), 19–38.
Google Scholar

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.
Google Scholar

K. Pichór and R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl. 249 (2000), 668–685.
Google Scholar

K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic operators and semigroups, J. Math. Anal. Appl. 436 (2016), 305–321.
Google Scholar

K. Pichór and R. Rudnicki, Asymptotic decomposition of substochastic semigroups and applications, Stoch. Dyn. 18 (2018), 1850001, 18 pp.
Google Scholar

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.
Google Scholar

K. Pichór and R. Rudnicki, Applications of stochastic semigroups to cell cycle models, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), 2365–2381.
Google Scholar

K. Pichór and R. Rudnicki, Dynamics of antibody levels: asymptotic properties, Math. Methods Appl. Sci. 43 (2020), 10490–10499.
Google Scholar

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.
Google Scholar

K. Pichór and R. Rudnicki, Asymptotic properties of a general model of immune status, SIAM J. Appl. Math. 83 (2023), 172–193.
Google Scholar

R. Rudnicki, Invariant measures for the flow of a first order partial differential equation, Ergodic Theory Dynam. Systems 5 (1985), 437–443.
Google Scholar

R. Rudnicki, Asymptotic properties of the iterates of positive operators on C(X), Bull. Polish Acad. Sci. Math. 34 (1986), 181–187.
Google Scholar

R. Rudnicki, Strong ergodic properties of a first-order partial differential equation, J. Math. Anal. Appl. 133 (1988), 14–26.
Google Scholar

R. Rudnicki, On asymptotic stability and sweeping for Markov operators, Bull. Polish Acad. Sci. Math. 43 (1995), 245–262.
Google Scholar

R. Rudnicki, Chaos for some infinite-dimensional dynamical systems, Math. Methods Appl. Sci. 27 (2004), 723–738.
Google Scholar

R. Rudnicki, Chaoticity of the blood cell production system, Chaos 19 (2009), 043112, 6 pp.
Google Scholar

R. Rudnicki, Chaoticity and invariant measures for a cell population model, J. Math. Anal. Appl. 393 (2012), 151–165.
Google Scholar

R. Rudnicki, An ergodic theory approach to chaos, Discrete Contin. Dyn. Syst. 35 (2015), 757–770.
Google Scholar

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.)
Google Scholar

R. Rudnicki, Ergodic properties of a semilinear partial differential equation, J. Differential Equations 372 (2023), 235–253.
Google Scholar

R. Rudnicki and A. Tomski, On a stochastic gene expression with pre-mRNA, mRNA and protein contribution, J. Theoret. Biol. 387 (2015), 54–67.
Google Scholar

R. Rudnicki and M. Tyran-Kamińska, Piecewise Deterministic Processes in Biological Models, SpringerBriefs Appl. Sci. Technol., Math. Methods, Springer, Cham, 2017.
Google Scholar

R. Rudnicki and P. Zwoleński, Model of phenotypic evolution in hermaphroditic populations, J. Math. Biol. 70 (2015), 1295–1321.
Google Scholar

Vol. 38 No. 2 (2024)
Published: 2024-07-23

ISSN: 0860-2107

eISSN: 2391-4238

10.1515/amsil

Publisher

University of Silesia Press

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