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：经典的可逆马尔可夫链同构定理在研究强对称马尔可夫过程的局部时间过程性质~\cite{mr06}、界定图上随机游走的覆盖时间~\cite{dlp11}，以及随机游走环路汤和布朗环路汤等物理学相关主题~\cite{lt07}中发挥了重要作用。Le Jan、Eisenbaum 和 Kaspi~\cite{lejan08, ek09, eisenbaum13} 发现了这些定理的非可逆版本。本文基于密度公式为所有这些非可逆同构定理提供了证明，推广了 \cite{bhs21} 中的结果。此外，我们运用该方法推广了 \cite{eisenbaum13} 中关于积和式过程的比较不等式，并推导出非可逆马尔可夫链覆盖时间的上界。