Quadratic stochastic operators and processes: Results and open problems

Rasul Ganikhodzhaev; Farrukh Mukhamedov; Utkir Rozikov

doi:10.1142/S0219025711004365

Quadratic stochastic operators and processes: Results and open problems

Rasul Ganikhodzhaev
, Farrukh Mukhamedov
, Utkir Rozikov

Research output: Contribution to journal › Article › peer-review

251 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	279-335
Number of pages	57
Journal	Infinite Dimensional Analysis, Quantum Probability and Related Topics
Volume	14
Issue number	2
DOIs	https://doi.org/10.1142/S0219025711004365
Publication status	Published - Jun 2011
Externally published	Yes

Keywords

Quadratic stochastic operator
Volterra and non-Volterra operators
ergodic
fixed point
quadratic stochastic process
quantum quadratic stochastic operator
quantum quadratic stochastic process
simplex
trajectory

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Statistics and Probability
Mathematical Physics
Applied Mathematics

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title = "Quadratic stochastic operators and processes: Results and open problems",

abstract = "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.",

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author = "Rasul Ganikhodzhaev and Farrukh Mukhamedov and Utkir Rozikov",

note = "Funding Information: This work was done within the scheme of Junior Associate at the ICTP, Trieste, Italy, and F.M. and U.R. thank ICTP for providing financial support and all facilities for his several visits to ICTP during 2005–2010. U.R. also supported by TWAS Research Grant No.: 09-009 RG/MATHS/AS−I–UNESCO FR:3240230333. The authors (R.G. and F.M.) acknowledge the MOSTI grants 01-01-08-SF0079 and CLB10-04. We are grateful to both referees for their suggestions which improved the style and contents of the paper.",

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N2 - 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.

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Quadratic stochastic operators and processes: Results and open problems

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Keywords

ASJC Scopus subject areas

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