Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth-death dynamics. We improve results in previous works [51,57] and provide weaker hypotheses under which 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 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 $\Gamma$-convergence of gradient flows, we show that on the torus, smooth and bounded positive solutions of the kernelized 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 minimizers of the energy corresponding to the kernelized dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernelized dynamics towards the Gibbs measure.

翻译：受非凸势能下吉布斯测度抽样挑战的启发，我们研究了一种连续型出生-死亡动力学。我们改进了先前工作[51,57]中的结论，在更弱的假设条件下证明了：由Kullback-Leibler散度或χ²散度支配的出生-死亡动力学概率密度，能够以与势垒无关的通用速率指数快速收敛至吉布斯平衡测度。为构建基于纯出生-死亡动力学的实用数值采样器，我们考虑了一种相互作用粒子系统——该系统受梯度流结构和经典Fokker-Planck方程启发，并依赖于基于核的测度近似。利用梯度流Γ-收敛技术，我们证明了在环面上，核化动力学光滑有界正解在有限时间区间内随着核带宽趋零而收敛至纯出生-死亡动力学。此外，我们给出了核化动力学对应能量极小化偏差的定量估计。最后，我们证明了核化动力学渐近态收敛至吉布斯测度的长时间渐近结果。