In this paper, we study the stability problem of a stochastic, nonlinear,
discrete-time system. We introduce a linear transfer operator-based Lyapunov
measure as a new tool for stability verification of stochastic systems. Weaker
set-theoretic notion of almost everywhere stochastic stability is introduced
and verified, using Lyapunov measure-based stochastic stability theorems.
Furthermore, connection between Lyapunov functions, a popular tool for
stochastic stability verification, and Lyapunov measures is established. Using
the duality property between the linear transfer Perron-Frobenius and Koopman
operators, we show the Lyapunov measure and Lyapunov function used for the
verification of stochastic stability are dual to each other. Set-oriented
numerical methods are proposed for the finite dimensional approximation of the
Perron-Frobenius operator; hence, Lyapunov measure is proposed. Stability
results in finite dimensional approximation space are also presented. Finite
dimensional approximation is shown to introduce further weaker notion of
stability referred to as coarse stochastic stability. The results in this paper
extend our earlier work on the use of Lyapunov measures for almost everywhere
stability verification of deterministic dynamical systems ("Lyapunov Measure
for Almost Everywhere Stability", {\it IEEE Trans. on Automatic Control}, Vol.
53, No. 1, Feb. 2008).

本文在研究中引入基于线性转移算子的李亚普诺夫测度作为随机系统稳定性验证的新工具，并建立了李亚普诺夫函数和测度之间的联系以及引入了用于有限维近似的集合定向数值方法，给出了有限维空间中的稳定性结果。

离散时间系统的随机稳定性分析：利用李亚普诺夫测度

Stochastic Stability Analysis of Discrete Time System Using Lyapunov  Measure

One of the proposed solutions to the equilibrium selection problem for agents
learning in repeated games is obtained via the notion of stochastic stability.
Learning algorithms are perturbed so that the Markov chain underlying the
learning dynamics is necessarily irreducible and yields a unique stable
distribution. The stochastically stable distribution is the limit of these
stable distributions as the perturbation rate tends to zero. We present the
first exact algorithm for computing the stochastically stable distribution of a
Markov chain. We use our algorithm to predict the long-term dynamics of simple
learning algorithms in sample repeated games.

用随机稳定性的概念可以解决代理在反复博弈中平衡选择问题，本文提出了一种精确计算马尔可夫链中随机稳定分布的算法，并将其用于预测简单学习算法在样本反复博弈中的长期动态。