On cubic stochastic operators and processes

TL;DR

This paper introduces cubic stochastic operators and processes on probability measures, extending previous quadratic models to continuous sets, and studies their dynamics through finite-dimensional trajectories and differential equations.

Contribution

It defines cubic stochastic operators and processes on continuous spaces, providing a framework for analyzing their dynamics via finite-dimensional simplifications and differential equations.

Findings

Construction of cubic stochastic operators on continuous spaces

Reduction of dynamics to finite-dimensional simplex trajectories

Derivation of differential equations for continuous-time processes

Abstract

In this paper analogically as quadratic stochastic operators and processes we define cubic stochastic operator (CSO) and cubic stochastic processes (CSP). These are defined on the set of all probability measures of a measurable space. The measurable space can be given on a finite or continual set. The finite case has been investigated before. So here we mainly work on the continual set. We give a construction of a CSO and show that dynamical systems generated by such a CSO can be studied by studying of the behavior of trajectories of a CSO given on a finite dimensional simplex. We define a CSP and drive differential equations for such CSPs with continuous time.