  We study convergence behaviors of degenerate and non-reversible stochastic
differential equations. Our method follows a Lyapunov method in probability
density space, in which the Lyapunov functional is chosen as a weighted
relative Fisher information functional. We construct a weighted Fisher
information induced Gamma calculus method with a structure condition. Under
this condition, an explicit algebraic tensor is derived to guarantee the
convergence rate for the probability density function converging to its
invariant distribution. We provide an analytical example for underdamped
Langevin dynamics with variable diffusion coefficients.
