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

Given a target distribution $\mu$ on a general state space $\mathcal{X}$ and a proposal Markov jump process with generator $Q$, the purpose of this paper is to investigate two universal properties enjoyed by two types of Metropolis-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 a geometric interpretation of $M_1$, $M_2$ and their convex combinations as $L^1$ minimizers between $Q$ and the set of $\mu$-reversible generators of Markov jump processes. Second, specializing into the case of $\mathcal{X} = \mathbb{R}^d$ along with a Gaussian proposal with vanishing variance and Gibbs target distribution, we prove that, upon appropriate scaling in time, the family of Markov jump processes corresponding to $M_1$, $M_2$ or their convex combinations all converge weakly to an universal Langevin diffusion. While $M_1$ and $M_2$ are seemingly different stochastic dynamics, it is perhaps surprising that they share these two universal properties. These two results are 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 are new.