TY - JOUR
T1 - Quadratic stochastic operators and processes
T2 - Results and open problems
AU - Ganikhodzhaev, Rasul
AU - Mukhamedov, Farrukh
AU - Rozikov, Utkir
N1 - 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.
PY - 2011/6
Y1 - 2011/6
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.
AB - 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.
KW - Quadratic stochastic operator
KW - Volterra and non-Volterra operators
KW - ergodic
KW - fixed point
KW - quadratic stochastic process
KW - quantum quadratic stochastic operator
KW - quantum quadratic stochastic process
KW - simplex
KW - trajectory
UR - http://www.scopus.com/inward/record.url?scp=79959767477&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=79959767477&partnerID=8YFLogxK
U2 - 10.1142/S0219025711004365
DO - 10.1142/S0219025711004365
M3 - Article
AN - SCOPUS:79959767477
SN - 0219-0257
VL - 14
SP - 279
EP - 335
JO - Infinite Dimensional Analysis, Quantum Probability and Related Topics
JF - Infinite Dimensional Analysis, Quantum Probability and Related Topics
IS - 2
ER -