Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics

Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth-death dynamics. We prove that the probability density of the birth-death governed by Kullback-Leibler divergence or by $\chi^2$ 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 relies on kernel-based approximations of the measure and retains the gradient-flow structure. We show on the torus that the kernelized dynamics $\Gamma$-converges, 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 minimizers of the energy corresponding to the kernalized dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernalized dynamics towards the Gibbs measure.