The classical isomorphism theorems for reversible Markov chains have played an important role in studying the properties of local time processes of strongly symmetric Markov processes~\cite{mr06}, bounding the cover time of a graph by a random walk~\cite{dlp11}, and in topics related to physics, such as random walk loop soups and Brownian loop soups~\cite{lt07}. Non-reversible versions of these theorems have been discovered by Le Jan, Eisenbaum, and Kaspi~\cite{lejan08, ek09, eisenbaum13}. Here, we give a density-formula-based proof for all these non-reversible isomorphism theorems, extending the results in \cite{bhs21}. Moreover, we use this method to generalize the comparison inequalities derived in \cite{eisenbaum13} for permanental processes and derive an upper bound for the cover time of non-reversible Markov chains.

翻译：经典可逆马尔可夫链的同构定理在以下方面发挥了重要作用：研究强对称马尔可夫过程的局部时性质~\cite{mr06}，通过随机游走限定图的覆盖时间~\cite{dlp11}，以及与物理相关的课题，例如随机游走环和布朗环~\cite{lt07}。Le Jan、Eisenbaum 和 Kaspi 发现了这些定理的非可逆版本~\cite{lejan08, ek09, eisenbaum13}。本文给出一个基于密度公式的证明，涵盖所有非可逆同构定理，并将 \cite{bhs21} 中的结果进行了推广。此外，我们利用此方法推广了 \cite{eisenbaum13} 中关于永久过程的不等式比较，并推导出了非可逆马尔可夫链覆盖时间的一个上界。