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Stochastic Operators and Semigroups and Their Applications in Physics and Biology

Part of the Lecture Notes in Mathematics book series (LNM,volume 2126)

Abstract

Stochastic operators are positive linear operators defined on the space of integrable functions preserving the set of densities. They appear in many fields of mathematics and applications and they are use to study ergodic properties of dynamical systems and describe the evolution of Markov chains. Stochastic semigroups are continuous semigroups of stochastic operators. They describe the behaviour of the distributions of Markov processes like diffusion processes, piecewise deterministic processes and hybrid stochastic processes. In this chapter we present many examples of stochastic operators and semigroups: The Frobenius–Perron operator, diffusion semigroups, flows semigroups with jumps and switching and semigroups related to hybrid systems. Then we present some results concerning their long-time behaviour: asymptotic stability, sweeping, completely mixing and convergence to self-similar solutions. The results concerning stochastic operators are applied to study ergodicity, mixing and exactness of dynamical systems and an integral operator appearing in the theory of cell cycle. The general results concerning stochastic semigroups are applied to diffusion processes, jump processes and biological models described by piecewise deterministic stochastic processes: birth-death processes, the evolution of the genome, gene expression and physiologically structured models.

Keywords

  • Asymptotic Stability
  • Iterate Function System
  • Invariant Density
  • Integral Semigroup
  • Stochastic Operator

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Acknowledgements

This research was partially supported by the State Committee for Scientific Research (Poland) Grant No. N N201 608240. The author is a supervisor in the International Ph.D. Projects Programme of Foundation for Polish Science operated within the Innovative Economy Operational Programme 2007–2013 (Ph.D. Programme: Mathematical Methods in Natural Sciences).

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Rudnicki, R. (2015). Stochastic Operators and Semigroups and Their Applications in Physics and Biology. In: Banasiak, J., Mokhtar-Kharroubi, M. (eds) Evolutionary Equations with Applications in Natural Sciences. Lecture Notes in Mathematics, vol 2126. Springer, Cham. https://doi.org/10.1007/978-3-319-11322-7_6

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