We provide a Lyapunov convergence analysis for time-inhomogeneous variable coefficient stochastic differential equations (SDEs). Three typical examples include overdamped, irreversible drift, and underdamped Langevin dynamics. We first formula the probability transition equation of Langevin dynamics as a modified gradient flow of the Kullback-Leibler divergence in the probability space with respect to time-dependent optimal transport metrics. This formulation contains both gradient and non-gradient directions depending on a class of time-dependent target distribution. We then select a time-dependent relative Fisher information functional as a Lyapunov functional. We develop a time-dependent Hessian matrix condition, which guarantees the convergence of the probability density function of the SDE. We verify the proposed conditions for several time-inhomogeneous Langevin dynamics. For the overdamped Langevin dynamics, we prove the $O(t^{-1/2})$ convergence in $L^1$ distance for the simulated annealing dynamics with a strongly convex potential function. For the irreversible drift Langevin dynamics, we prove an improved convergence towards the target distribution in an asymptotic regime. We also verify the convergence condition for the underdamped Langevin dynamics. Numerical examples demonstrate the convergence results for the time-dependent Langevin dynamics.

翻译：本文针对时间非齐次变系数随机微分方程（SDEs）提供了Lyapunov收敛性分析。三个典型例子包括过阻尼、不可逆漂移和欠阻尼Langevin动力学。我们首先将Langevin动力学的概率转移方程表述为概率空间中关于时间依赖最优传输度量的Kullback-Leibler散度的修正梯度流。这种表述包含依赖于一类时间依赖目标分布的梯度方向和非梯度方向。接着，我们选择时间依赖的相对Fisher信息泛函作为Lyapunov泛函。我们发展了一个时间依赖的Hessian矩阵条件，该条件保证了SDE概率密度函数的收敛性。我们验证了若干时间非齐次Langevin动力学的提出条件。对于过阻尼Langevin动力学，我们证明了具有强凸势函数的模拟退火动力学在$L^1$距离下的$O(t^{-1/2})$收敛性。对于不可逆漂移Langevin动力学，我们证明了在渐近区域中向目标分布的改进收敛性。我们还验证了欠阻尼Langevin动力学的收敛条件。数值算例展示了时间依赖Langevin动力学的收敛结果。