In this paper, we first introduce and define several new information divergences in the space of transition matrices of finite Markov chains which measure the discrepancy between two Markov chains. These divergences offer natural generalizations of classical information-theoretic divergences, such as the $f$-divergences and the R\'enyi divergence between probability measures, to the context of finite Markov chains. We begin by detailing and deriving fundamental properties of these divergences and notably gives a Markov chain version of the Pinsker's inequality and Chernoff information. We then utilize these notions in a few applications. First, we investigate the binary hypothesis testing problem of Markov chains, where the newly defined R\'enyi divergence between Markov chains and its geometric interpretation play an important role in the analysis. Second, we propose and analyze information-theoretic (Ces\`aro) mixing times and ergodicity coefficients, along with spectral bounds of these notions in the reversible setting. Examples of the random walk on the hypercube, as well as the connections between the critical height of the low-temperature Metropolis-Hastings chain and these proposed ergodicity coefficients, are highlighted.

翻译：本文首先在有限马尔可夫链转移矩阵空间中引入并定义了几种新的信息散度，用于度量两个马尔可夫链之间的差异。这些散度将经典信息论散度（如概率测度间的 $f$-散度和Rényi散度）自然推广至有限马尔可夫链语境。我们详细阐述了这些散度的基本性质并推导了相关结论，特别给出了马尔可夫链版本的Pinsker不等式和Chernoff信息。随后，我们将这些概念应用于若干场景。首先，研究了马尔可夫链的二元假设检验问题，其中新定义的马尔可夫链间Rényi散度及其几何解释在分析中起关键作用。其次，提出并分析了信息论（Cesàro）混合时间与遍历性系数，并在可逆设定下给出了这些概念的谱界。文中还以超立方体上的随机游走为例，并强调了低温Metropolis-Hastings链的临界高度与所提遍历性系数之间的联系。