Abstract

A linear stochastic (Markov) operator is a positive linear contraction which preserves the simplex. A quadratic stochastic (nonlinear Markov) operator is a positive symmetric bilinear operator that preserves the simplex. The ergodic theory studies the long term average behavior of systems evolving in time. The classical mean ergodic theorem asserts that the arithmetic average of the linear stochastic operator always converges to some linear stochastic operator. While studying the evolution of the population system, S. Ulam conjectured the mean ergodicity of quadratic stochastic operators. However, M. Zakharevich showed that Ulam’s conjecture is false in general. Later, N. Ganikhodjaev and D. Zanin have generalized Zakharevich’s example in the class of quadratic stochastic Volterra operators. Afterward, N. Ganikhodjaev made a conjecture that Ulam’s conjecture is true for properly quadratic stochastic non-Volterra operators. In this paper, we provide counterexamples to Ganikhodjaev’s conjecture on mean ergodicity of quadratic stochastic operators acting on the higher dimensional simplex.

中文翻译：

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Ganikhodjaev关于二次随机算子平均遍历性的猜想

摘要

线性随机（Markov）算子是保持单形的正线性收缩。二次随机（非线性马尔可夫）算子是保留单形的正对称双线性算子。遍历理论研究了随时间演变的系统的长期平均行为。经典的平均遍历定理认为，线性随机算子的算术平均值总是收敛于某个线性随机算子。在研究人口系统的演化过程中，S。Ulam推测了二次随机算子的平均遍历性。但是，扎克哈列维奇先生（M. Zakharevich）表明，乌兰姆的猜想总体上是错误的。后来，N。Ganikhodjaev和D.Zanin在二次随机Volterra算子类别中推广了Zakharevich的例子。之后，N。Ganikhodjaev猜想Ulam猜想对于二次随机非Volterra算子是正确的。在本文中，我们提供了Ganikhodjaev猜想的反例，该猜想是作用在高维单纯形上的二次随机算子的平均遍历性的。