Entropy Dissipation for Degenerate Stochastic Differential Equations via Sub-Riemannian Density Manifold

• Published in: Entropy (Basel, Switzerland) Vol. 25; no. 5; p. 786
• Main Authors: Feng, Qi, Li, Wuchen
• Format: Journal Article
• Language: English
• Published: Switzerland MDPI AG 11-05-2023; MDPI
• Subjects: Analysis; auxiliary Fisher information; Calculus; Convergence; degenerate drift–diffusion process; Density; Differential equations; Diffusion; Distribution (Probability theory); Divergence; Entropy; Fisher information; Formulas (mathematics); generalized Bochner’s formula; Lie groups; Lyapunov methods; Mathematical analysis; sub-Riemannian density manifold; auxiliary Fisher information; generalized Bochner’s formula; degenerate drift–diffusion process; Lyapunov methods; sub-Riemannian density manifold
• ISSN: 1099-4300; 1099-4300
• DOI: 10.3390/e25050786

We studied the dynamical behaviors of degenerate stochastic differential equations (SDEs). We selected an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conducted the Lyapunov exponential convergence analysis of degenerate SDEs. We derived the convergence rate condition by generalized Gamma calculus. Examples of the generalized Bochner's formula are provided in the Heisenberg group, displacement group, and Martinet sub-Riemannian structure. We show that the generalized Bochner's formula follows a generalized second-order calculus of Kullback-Leibler divergence in density space embedded with a sub-Riemannian-type optimal transport metric.