在本研究中，我们将在申请人及其合作者先前研究工作的基础上，探讨不可逆Markov链的Metropolis-Hastings（MH）可逆化。本研究的目的是进一步发展MH可逆化理论，其应用方向有两个：一是作为分析不可逆Markov链的工具，二是以其优越的混合表现作为一个理论上更为理想的算法替代传统MH算法。围绕这两个方向，我们提出的研究包括三个部分。首先，我们将提出并证明一个关于MH可逆性的Markov链树定理的推广。这可能使我们得到关于不可逆Markov链特征值的新估计和这类链收敛速度的新边界。其次，在MH可逆性的启发下，我们将研究模拟退火的一个加速变体。最后，我们将研究MH可逆性在空间和时间尺度上的scaling极限。这可能使我们得到关于MH可逆化的新的扩散近似，并对模拟这些Markov链具有实践指导意义。

In the proposed research, building upon our previous research work we will investigate the Metropolis-Hastings (MH) reversiblizations of non-reversible Markov chains. In particular, we will focus on further developing the theory of MH reversiblizations in two important directions: first as a tool for analyzing non-reversible Markov chains, and second as a theoretically promising alternative of the classical MH algorithm with superior mixing performance. Our proposed research consists of three parts around these two directions. First, we shall propose and prove a generalized version of Markov chain tree theorem for the MH reversiblizations. This may lead to new estimates on the eigenvalues and new bounds on the convergence rate of non-reversible Markov chains. Second, we will investigate an accelerated variant of simulated annealing inspired by the MH reversiblizations. Third, we will look into the scaling limit of the MH reversiblizations upon scaling in space and in time. This may lead to new diffusion approximation of the MH reversiblizations and give practical insights on simulating these Markov chains.