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We study the dynamical behaviors of degenerate stochastic differentialequations (SDEs). We select an auxiliary Fisher information functional as theLyapunov functional. Using generalized Fisher information, we conduct theLyapunov exponential convergence analysis of degenerate SDEs. We derive theconvergence rate condition by generalized Gamma calculus. Examples of thegeneralized Bochner's formula are provided in the Heisenberg group,displacement group, and Martinet sub-Riemannian structure. We show that thegeneralized Bochner's formula follows a generalized second-order calculus ofKullback-Leibler divergence in density space embedded with a sub-Riemanniantype optimal transport metric.

Entropy dissipation for degenerate stochastic differential equations via sub-Riemannian density manifold