QUADRATIC STOCHASTIC OPERATORS AND PROCESSES: RESULTS AND OPEN PROBLEMS
Top Cited Papers
- 1 June 2011
- journal article
- review article
- Published by World Scientific Pub Co Pte Ltd in Infinite Dimensional Analysis, Quantum Probability and Related Topics
- Vol. 14 (02), 279-335
- https://doi.org/10.1142/s0219025711004365
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.Keywords
This publication has 46 references indexed in Scilit:
- On l-Volterra quadratic stochastic operatorsDoklady Mathematics, 2009
- F-quadratic stochastic operatorsMathematical Notes, 2008
- On expansion of quantum quadratic stochastic processes into fibrewise Markov processes defined on von Neumann algebrasIzvestiya: Mathematics, 2004
- On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplexRussian Mathematical Surveys, 2004
- On uniform ergodic theorems for quadratic processes onC*-algebrasSbornik: Mathematics, 2000
- Infinite-dimensional quadratic Volterra operatorsRussian Mathematical Surveys, 2000
- Ergodic properties of discrete quadratic stochastic processes defined on von Neumann algebrasIzvestiya: Mathematics, 2000
- Ergodic properties of quantum quadratic stochastic processes defined on von Neumann algebrasRussian Mathematical Surveys, 1998
- On the definition of bistochastic quadratic operatorsRussian Mathematical Surveys, 1993
- Compact matrix pseudogroupsCommunications in Mathematical Physics, 1987