Foundations of locally-balanced Markov processes

We formally introduce and study locally-balanced Markov jump processes (LBMJPs) defined on a general state space. These continuous-time stochastic processes with a user-specified limiting distribution are designed for sampling in settings involving discrete parameters and/or non-smooth distributions, addressing limitations of other processes such as the overdamped Langevin diffusion. The paper establishes the well-posedness, non-explosivity, and ergodicity of LBMJPs under mild conditions. We further explore regularity properties such as the Feller property and characterise the weak generator of the process. We then derive conditions for exponential ergodicity via spectral gaps and establish comparison theorems for different balancing functions. In particular we show an equivalence between the spectral gaps of Metropolis--Hastings algorithms and LBMJPs with bounded balancing function, but show that LBMJPs can exhibit uniform ergodicity on unbounded state spaces when the balancing function is unbounded, even when the limiting distribution is not sub-Gaussian. We also establish a diffusion limit for an LBMJP in the small jump limit, and discuss applications to Monte Carlo sampling and non-reversible extensions of the processes.