Quadratic stochastic operators and processes: Results and open problems

Rasul Ganikhodzhaev, Farrukh Mukhamedov, Utkir Rozikov

Research output: Contribution to journalArticlepeer-review

206 Citations (Scopus)


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 languageEnglish
Pages (from-to)279-335
Number of pages57
JournalInfinite Dimensional Analysis, Quantum Probability and Related Topics
Issue number2
Publication statusPublished - Jun 2011
Externally publishedYes


  • 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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