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Given a target distribution $\mu$ on a general state space $\mathcal{X}$ anda proposal Markov jump process with generator $Q$, the purpose of this paper isto investigate two universal properties enjoyed by two types ofMetropolis-Hastings (MH) processes with generators $M_1(Q,\mu)$ and$M_2(Q,\mu)$ respectively. First, we motivate our study of $M_2$ by offering ageometric interpretation of $M_1$, $M_2$ and their convex combinations as $L^1$minimizers between $Q$ and the set of $\mu$-reversible generators of Markovjump processes. Second, specializing into the case of $\mathcal{X} =\mathbb{R}^d$ along with a Gaussian proposal with vanishing variance and Gibbstarget distribution, we prove that, upon appropriate scaling in time, thefamily of Markov jump processes corresponding to $M_1$, $M_2$ or their convexcombinations all converge weakly to an universal Langevin diffusion. While$M_1$ and $M_2$ are seemingly different stochastic dynamics, it is perhapssurprising that they share these two universal properties. These two resultsare known for $M_1$ in Billera and Diaconis (2001) and Gelfand and Mitter(1991), and the counterpart results for $M_2$ and their convex combinations arenew.

Universality of the Langevin diffusion as scaling limit of a family of Metropolis-Hastings processes I: fixed dimension