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

TL;DR

This paper investigates the entropy dissipation and convergence properties of degenerate stochastic differential equations using sub-Riemannian geometry and generalized Fisher information, providing new analytical tools and examples.

Contribution

It introduces a novel Lyapunov functional based on Fisher information and derives convergence rates for degenerate SDEs using generalized Gamma calculus.

Findings

Established exponential convergence of degenerate SDEs via generalized Fisher information

Derived generalized Bochner's formula in sub-Riemannian structures

Provided examples in Heisenberg, displacement, and Martinet groups

Abstract

We study the dynamical behaviors of degenerate stochastic differential equations (SDEs). We select an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conduct the Lyapunov exponential convergence analysis of degenerate SDEs. We derive 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.