On some properties of quadratic stochastic processes (original) (raw)

Quadratic Stochastic Operators and Processes: Results and Open Problems

Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2011

The history of the quadratic stochastic operators can be traced back to the work of Bernshtein (1924). For more than 80 years, this theory has been developed and many papers were published. In recent years it has again become of interest in connection with its numerous applications in many branches of mathematics, biology and physics. But most results of the theory were published in non-English journals, full text of which are not accessible. In this paper we give all necessary definitions and a brief description of the results for three cases: (i) discrete-time dynamical systems generated by quadratic stochastic operators; (ii) continuous-time stochastic processes generated by quadratic operators; (iii) quantum quadratic stochastic operators and processes. Moreover, we discuss several open problems.

On quadratic stochastic processes and related differential equations

Journal of Physics: Conference Series, 2013

It is known that the theory of Markov process is a rapidly developing field with numerous applications to many branches of mathematics and physics, biology and so on. But there are some physical models which cannot be described by such processes. One of such models is related to population genetics. These processes are called quadratic stochastic processes (q.s.p.). In this theory it is important to construct nontrivial examples of such processes. In the present paper we are going to provide a construction of q.s.p. by means of two given processes. We should stress that such a construction allows us to produce lots of nontrivial examples of q.s.o. We also associate to given q.s.p. two kind of processes. Note that one of such processes is Markov. It is proved that such kind of processes uniquely define q.s.p. Moreover, we also derive some differential equations for q.s.p.

The Stability of some stochastic processes

2010

We formulate and prove a new criterion for stability of e-processes. It says that any e-process which is averagely bounded and concentrating is asymptotically stable. In the second part, we show how this general result applies to some shell models (the Goy and the Sabra model). Indeed, we manage to prove that the processes corresponding to these models satisfy the

ON DYNAMICS OF ξ S QUADRATIC STOCHASTIC OPERATORS

International Journal of Modern Physics: Conference Series, 2012

In this research we introduce a new class of quadratic stochastic operators called ξ s -QSO which are defined through coefficient of the operator from measure-theoretic (namely we are looking the coefficient as the measures which are absolute continuous or singular) point of view. We also study the limiting behaviour of ξ s -QSO defined on 2D-simplex. We first describe ξ s -QSO on 2Dsimplex and classify them with respect to the conjugacy and renumeration of the coordinates. We find six non-isomorphic classes of such operators. Moreover, we investigate the behaviour of each operator from three classes and prove convergence of trajectories of these classes and study their certain properties. We showed trajectories of two classes converge to the equilibrium. For the third class, it is established only the negative trajectories converge to the equilibrium.

On nonhomogeneous geometric quadratic stochastic operators

Turkish Journal of Mathematics

In this paper, we construct a nonhomogeneous geometric quadratic stochastic operator generated by 2partition ξ on countable state space X = Z *. The limiting behavior of such operator is studied. We have proved that such operator possesses the regular property.

On Quadratic Stochastic Operators Having Three Fixed Points

Journal of Physics: Conference Series, 2016

We knew that a trajectory of a linear stochastic operator associated with a positive square stochastic matrix starting from any initial point from the simplex converges to a unique fixed point. However, in general, the similar result for a quadratic stochastic operator associated with a positive cubic stochastic matrix does not hold true. In this paper, we provide an example for the quadratic stochastic operator with positive coefficients in which its trajectory may converge to different fixed points depending on initial points.

On a class of separable quadratic stochastic operators

Lobachevskii Journal of Mathematics, 2011

The purpose of this paper is to investigate a class of separable quadratic stochastic operators. Each separable quadratic stochastic operator (SQSO) depends on two quadratic matrices A and B, which have some relations. In this paper we proved that for each skew symmetric matrix A the corresponding SQSO is a linear operator. We also proved that non linear Volterra QSOs are not SQSOs. For a fixed matrix A we also discussed some properties of the set of all the corresponding matrices B of SQSOs.

Stability in distribution of randomly perturbed quadratic maps as Markov processes

The Annals of Applied Probability, 2004

Iteration of randomly chosen quadratic maps defines a Markov process: Xn+1 = εn+1Xn(1 − Xn), where εn are i.i.d. with values in the parameter space [0, 4] of quadratic maps F θ (x) = θx(1 − x). Its study is of significance as an important Markov model, with applications to problems of optimization under uncertainty arising in economics. In this article a broad criterion is established for positive Harris recurrence of Xn.