The Density Formula Approach for Non-reversible Isomorphism Theorems, with Applications
Qinghua, Ding, Venkat Anantharam
TL;DR
This work develops a density-formula framework for 1-permanental vectors with kernels whose symmetric part is positive definite and uses Ward identities to provide unified, non-reversible proofs of Dynkin's, Ray-Knight's, and Eisenbaum's isomorphism theorems. By coupling these density formulas with interpolation arguments, it derives Kahane-type and Slepian-type comparison inequalities for 1-permanental processes and extends these tools to the non-reversible setting. The approach yields a symmetrization-based bound on the cover time of non-reversible Markov chains, showing $t_{\text{cov}}=O\big(t_{\text{cov}}' \log \gamma\big)$, matching reversible behavior up to logarithmic factors in typical regimes. The results connect loop-soup local times and twisted Gaussian densities as a non-reversible analogue of Gaussian free fields, offering a versatile methodology for analyzing non-symmetric Markov processes and permanental structures.
Abstract
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.