Fisher information dissipation for time inhomogeneous stochastic
  differential equations
Qi Feng Xinzhe Zuo Wuchen Li
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.