Journal
INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS
Volume 14, Issue 2, Pages 279-335Publisher
WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0219025711004365
Keywords
Quadratic stochastic operator; quadratic stochastic process; quantum quadratic stochastic operator; quantum quadratic stochastic process; fixed point; trajectory; Volterra and non-Volterra operators; ergodic; simplex
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Funding
- ICTP
- TWAS [09-009 RG/MATHS/AS_I-UNESCO FR:3240230333]
- MOSTI [01-01-08-SF0079, CLB10-04]
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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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