Ahmad Fadillah Embong

On non-linear Markov operators: surjectivity vs orthogonal preserving property

Abstract In the present paper, we consider non-linear Markov operators, namely polynomial stochastic operators. We introduce a notion of orthogonal preserving polynomial stochastic operators. The purpose of this study is to show that surjectivity of non-linear Markov operators is equivalent to their orthogonal preserving property.

On stable b-bistochastic quadratic stochastic operators and associated non-homogenous Markov chains

In the present paper, we consider a class of quadratic stochastic operators (q.s.o.) called b-bistochastic q.s.o. defined on a finite dimensional simplex. We include several properties of b-bistochastic q.s.o. and their dynamical behaviour. One of the main findings in this paper is the description on the uniqueness of the fixed points. Besides, we list the conditions on strict contractive b-bistochastic q.s.o. on low-dimensional simplices, and it turns out that, the uniqueness of the fixed point does not imply its strict contractivity. Finally, we associate non-homogeneous Markov measures with b-bistochastic q.s.o. The defined measures were proven to satisfy the mixing property for regular b-bistochastic q.s.o. Moreover, we show that non-homogeneous Markov measures associated with a class of b-bistochastic q.s.o on one-dimensional simplex, meet the absolute continuity property.

Orthogonal preserving quadratic stochastic operators: Infinite dimensional case

In the present paper, we consider orthogonal preserving quadratic stochastic operators defined on infinite dimensional simplex. We provide a full descriptions of such kind of operators. Moreover, certain examples are given.

b— Bistochastic Quadratic Stochastic Operators and Their Properties

In the present paper, we consider a class of quadratic stochastic operators (q.s.o.) called 6-bistochastic q.s.o. We study several descriptive properties of b- bistochastic q.s.o. It turns out that, upper triangular stochastic matrix defines a linear b-stochastic operator. This allowed us to find some sufficient conditions on cubic stochastic matrix to be a b—bistochastic q.s.o.

On mixing of Markov measures associated with b−bistochastic QSOs

New majorization is in advantage as compared to the classical one since it can be defined as a partial order on sequences. We call it as b−order. Further, the defined order is used to establish a bistochasticity of nonlinear operators in which, in this study is restricted to the simplest case of nonlinear operators i.e quadratic operators. The discussions in this paper are based on bistochasticity of Quadratic Stochastic Operators (QSO) with respect to the b−order. In short, such operators are called b−bistochastic QSO. The main objectives in this paper are to show the construction of non-homogeneous Markov measures associated with QSO and to show the defined measures associated with the classes of b−bistochastic QSOs meet the mixing property.