We introduce the concept of an imprecise Markov semigroup $\mathbf{Q}$. It is a tool that allows to represent ambiguity around both the initial and the transition probabilities of a Markov process via a compact collection of plausible Markov semigroups, each associated with a (different, plausible) Markov process. We use techniques from geometry, functional analysis, and (high dimensional) probability to study the ergodic behavior of $\mathbf{Q}$. We show that, if the initial distribution of the Markov processes associated with the elements of $\mathbf{Q}$ is known and invariant, under some conditions that also involve the geometry of the state space, eventually the ambiguity around their transition probability fades. We call this property ergodicity of the imprecise Markov semigroup, and we relate it to the classical notion of ergodicity. We prove ergodicity both when the state space is Euclidean or a Riemannian manifold, and when it is an arbitrary measurable space. The importance of our findings for the fields of machine learning and computer vision is also discussed.

翻译：我们引入了不精确马尔可夫半群 $\mathbf{Q}$ 的概念。该工具能够通过一个紧致的、包含多个合理马尔可夫半群的集合来表示马尔可夫过程在初始概率与转移概率上的模糊性，其中每个半群均与一个（不同的、合理的）马尔可夫过程相关联。我们运用几何学、泛函分析以及（高维）概率论中的技术来研究 $\mathbf{Q}$ 的遍历行为。研究表明，若与 $\mathbf{Q}$ 中各元素相关联的马尔可夫过程的初始分布已知且不变，并在某些涉及状态空间几何结构的条件下，其转移概率的模糊性最终会消失。我们将此性质称为不精确马尔可夫半群的遍历性，并将其与经典的遍历性概念相联系。我们分别在状态空间为欧几里得空间或黎曼流形，以及为任意可测空间的情形下证明了遍历性。本文还探讨了我们的研究结果在机器学习与计算机视觉领域的重要意义。