Main goal of this paper is to formulate possibly simple and easy to verify criteria on existence of the unique attracting probability measure for stochastic process induced by generalized iterated function systems with probabilities (GIFSPs). To do this, we study the long-time behavior of trajectories of Markov-type operators acting on product of spaces of Borel measures on arbitrary Polish space. Precisely, we get the desired geometric rate of convergence of sequences of measures under the action of such operator to the unique distribution in the Hutchinson–Wasserstein distance. We apply the obtained results to study limiting behavior of random trajectories of GIFSPs as well as stochastic difference equations with multiple delays.

中文翻译：

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关于具有概率的广义 IFS 的几何遍历性

本文的主要目标是制定可能简单且易于验证的标准，以证明由具有概率的广义迭代函数系统（GIFSP）引起的随机过程的唯一吸引概率测度是否存在。为此，我们研究了马尔可夫型算子的轨迹在任意波兰空间上作用于 Borel 度量空间的乘积的长期行为。准确地说，我们得到了在这种算子的作用下，度量序列对 Hutchinson-Wasserstein 距离中唯一分布的期望几何收敛速度。我们将获得的结果应用于研究 GIFSP 随机轨迹的限制行为以及具有多个延迟的随机差分方程。