On Volterra quadratic stochastic operators with continual state space
Let (X,F) be a measurable space, and S(X,F) be the set of all probability measures on (X,F) where X is a state space and F is σ - algebraon X. We consider a nonlinear transformation (quadratic stochastic operator) defined by (Vλ)(A) = ∫{sub X}∫{sub X}P(x,y,A)dλ(x)dλ(y), where P(x, y, A) is regarded...
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| creator | Ganikhodjaev, Nasir Hamzah, Nur Zatul Akmar |
| description | Let (X,F) be a measurable space, and S(X,F) be the set of all probability measures on (X,F) where X is a state space and F is σ - algebraon X. We consider a nonlinear transformation (quadratic stochastic operator) defined by (Vλ)(A) = ∫{sub X}∫{sub X}P(x,y,A)dλ(x)dλ(y), where P(x, y, A) is regarded as a function of two variables x and y with fixed A ∈ F . A quadratic stochastic operator V is called a regular, if for any initial measure the strong limit lim{sub n→∞} V{sup n }(λ) is exists. In this paper, we construct a family of quadratic stochastic operators defined on the segment X = [0,1] with Borel σ - algebra F on X , prove their regularity and show that the limit measure is a Dirac measure. |
| doi_str_mv | 10.1063/1.4915658 |
| format | Conference Proceeding |
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| identifier | ISSN: 0094-243X |
| ispartof | AIP conference proceedings, 2015, Vol.1660 (1) |
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| source | AIP Journals Complete |
| subjects | ALGEBRA CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS FUNCTIONS MATHEMATICAL METHODS AND COMPUTING MATHEMATICAL OPERATORS MATHEMATICAL SOLUTIONS NONLINEAR PROBLEMS PROBABILITY SPACE STOCHASTIC PROCESSES TRANSFORMATIONS |
| title | On Volterra quadratic stochastic operators with continual state space |
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