PhD dissertation (in Polish): Ergodyczne w{\l}asno\'sci pewnych stochastycznych uk{\l}ad\'ow dynamicznych

TL;DR

This dissertation studies the ergodic properties of certain stochastic dynamical systems, including a cell division model, establishing unique invariant measures, stability, convergence rates, and limit theorems, with implications for biology.

Contribution

It proves the existence of a unique invariant measure and analyzes the ergodic behavior of a cell division model within stochastic dynamical systems.

Findings

Existence of a unique invariant measure for the model

Asymptotic stability of the model established

Convergence rate of measures to the invariant measure evaluated

Abstract

The dissertation describes ergodic properties of some stochastic dynamical systems generated by Markov chains with values in the state space which is a Polish space. The mathematical model describing the process of cell division is analyzed. The assumptions are satisfied, among others, by the model introduced by J.J. Tyson and K.B. Hannsgen, J. Math. Biol. (1986). Within the thesis the existence of a unique invariant measure is established and asymptotic stability of the model is verified. In addition, the rate of convergence of the sequence of measures, transformed by the subsequent iterations of Markov operator, to the unique invariant measure is evaluated. Further investigation of the model (in terms of its ergodic properties) concerns establishing the proofs of the central limit theorem and the law of the iterated logarithm. The results presented in the dissertation solve interesting mathematical problems, but, since they are inspired by the biological process of cell division, they may be also important for biology.