Ganikhodjaev’s Conjecture on Mean Ergodicity of Quadratic Stochastic Operators

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.