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Quadratic stochastic operators and processes: results and open problems. (English) Zbl 1242.60067
This paper reviews the modern developments of the theory and applications of quadratic stochastic operators initiated by S. Bernstein in 1924, based in large part on the literature published by the authors in non-English journals over the past 20 years. This broad overview is motivated by the use of quadratic stochastic operators in discrete-time population genetics, and more generally it includes the connections of quadratic operators and processes in classical, and quantum, probability with biology, genetics, population dynamics, and physics. The paper covers discrete dynamical systems and continuous-time stochastic processes generated by quadratic stochastic operators, as well as quantum quadratic stochastic operators and associated processes.

60H25Random operators and equations
60G07General theory of stochastic processes
37N25Dynamical systems in biology
46L53Noncommutative probability and statistics
60J28Applications of continuous-time Markov processes on discrete state spaces
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