Birth–death dynamics for sampling: global convergence, approximations and their asymptotics

Abstract

Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth–death dynamics. We improve results in previous works (Liu et al 2023 Appl. Math. Optim. 87 48; Lu et al 2019 arXiv:1905.09863) and provide weaker hypotheses under which the probability density of the birth–death governed by Kullback–Leibler divergence or by χ 2 divergence converge exponentially fast to the Gibbs equilibrium measure, with a universal rate that is independent of the potential barrier. To build a practical numerical sampler based on the pure birth–death dynamics, we consider an interacting particle system, which is inspired by the gradient flow structure and the classical Fokker–Planck equation and relies on kernel-based approximations of the measure. Using the technique of Γ-convergence of gradient flows, we show that on the torus, smooth and bounded positive solutions of the kernelised dynamics converge on finite time intervals, to the pure birth–death dynamics as the kernel bandwidth shrinks to zero. Moreover we provide quantitative estimates on the bias of minimisers of the energy corresponding to the kernelised dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernelised dynamics towards the Gibbs measure.