Generalized Gamma z calculus

Generalized Gamma z calculus

A drift-diffusion process with a non-degenerate diffusion coefficient matrix possesses good properties: convergence to equilibrium, entropy dissipation rate, etc. The degenerate drift-diffusion possesses a non-positive definite diffusion coefficient matrix, which makes it difficult to govern the convergence property and entropy dissipation rate by drift-diffusion coefficients on its own because of lacking control for the system. In general, the degenerate drift-diffusion is intrinsically equipped with a sub-Riemannian structure defined by the diffusion coefficients. We propose a new methodology to systematically study the general drift-diffusion process through sub-Riemannian geometry and Wasserstein geometry. We generalize the Bakry-Emery calculus and Gamma z calculus to define a new notion of sub-Riemannian Ricci curvature tensor. With the new Ricci curvature tensor, we are able to establish generalized curvature dimension bounds on sub-Riemannian manifolds which goes beyond step two condition. As an application, for the first time, we establish analytical bounds for logarithmic Sobolev inequalities for the weighted measure on the displacement group and Engel group. Our result also provides an entropy dissipation rate for Langevin dynamics with gradient drift and variable temperature matrix. The talk is based on joint works with Qi Feng.

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Wuchen Li

May 06, 2020
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