Given a target distribution $\pi$ and an arbitrary Markov infinitesimal generator $L$ on a finite state space $\mathcal{X}$, we develop three structured and inter-related approaches to generate new reversiblizations from $L$. The first approach hinges on a geometric perspective, in which we view reversiblizations as projections onto the space of $\pi$-reversible generators under suitable information divergences such as $f$-divergences. With different choices of functions $f$, we not only recover nearly all established reversiblizations but also unravel and generate new reversiblizations. Along the way, we unveil interesting geometric results such as bisection properties, Pythagorean identities, parallelogram laws and a Markov chain counterpart of the arithmetic-geometric-harmonic mean inequality governing these reversiblizations. This further serves as motivation for introducing the notion of information centroids of a sequence of Markov chains and to give conditions for their existence and uniqueness. Building upon the first approach, we view reversiblizations as generalized means. In this second approach, we construct new reversiblizations via different natural notions of generalized means such as the Cauchy mean or the dual mean. In the third approach, we combine the recently introduced locally-balanced Markov processes framework and the notion of convex $*$-conjugate in the study of $f$-divergence. The latter offers a rich source of balancing functions to generate new reversiblizations.

翻译：针对给定目标分布$\pi$及有限状态空间$\mathcal{X}$上的任意马尔可夫无穷小生成元$L$，本文提出三种相互关联的结构化方法来生成$L$的新可逆化表示。第一种方法基于几何视角，将可逆化视为在$f$-散度等合适信息散度下对$\pi$-可逆生成元空间的正交投影。通过选择不同的函数$f$，我们不仅能复现几乎所有已知的可逆化形式，还可揭示并生成新的可逆化结构。在此过程中，我们发现了有趣的几何性质，例如这些可逆化所遵循的平分性质、勾股恒等式、平行四边形法则以及算术-几何-调和均值不等式的马尔可夫链对应形式。该发现进一步激发我们引入马尔可夫链序列的信息质心概念，并给出其存在唯一性条件。基于第一种方法，我们将可逆化视为广义均值。在第二种方法中，我们通过柯西均值或对偶均值等不同的广义均值自然概念构建新的可逆化形式。在第三种方法中，我们结合最近提出的局部平衡马尔可夫过程框架与$f$-散度研究中的凸共轭概念，后者为生成新可逆化提供了丰富的平衡函数来源。