Quadratic stochastic operator on infinite state space: some rigorous and computational results

Abstract

The theory of quadratic stochastic operator (QSO) on finite state space is well developed to the study of the limiting behaviour and ergodic properties of trajectories of QSO. Thus, it motivates us to extend and introduce the theory of QSO on infinite state space. In this thesis, we constructed some new classes of quadratic stochastic operator on infinite state space, namely, Geometric QSO, Poisson QSO, Lebesgue QSO and Gaussian QSO. Moreover, we provide some analytical proof and computational numerical analysis to show the ergodicity and regularity of the constructed QSOs. It is shown that the Geometric QSO, Poisson QSO, and Lebesgue QSO are regular and ergodic. While, for Gaussian QSO, it is shown that they are regular for some values of parameters and nonregular for other values of parameters.