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Diffusion hypercontractivity via generalized density manifold

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Abstract

We prove a one-parameter family of diffusion hypercontractivity from a class of drift-diffusion processes. We next derive the related log–Sobolev, Poincare, and Talagrand inequalities. The derivation is based on the calculation of Hessian operators along generalized gradient flows in Dolbeault–Nazaret–Savare metric spaces (Dolbeault et al., Calc Var Partial Differ 2:193–231, 2010). In this direction, a mean-field type Bakry–Emery iterative calculus is presented. In particular, an inequality among Pearson divergence (P), negative Sobolev metric (\(H^{-1}\)), and generalized Fisher information functional (I), named \(PH^{-1}I\) inequality, is presented.

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Notes

  1. It is often named \(\alpha \)-divergence with \(\gamma =\frac{3-\alpha }{2}\). We use notation \(\gamma \) for the simplicity of presentation.

  2. When \(\gamma =1\), we remark that the notation of \({\mathcal {W}}_{1}\) represents the classical \(L^2\)-Wasserstein distance, not the \(L^1\)-Wasserstein distance.

Abbreviations

M :

Base manifold

g, \((\cdot ,\cdot )\) :

Metric

\(\Vert \cdot \Vert \) :

Norm

\(\nabla \cdot \) :

Divergence operator

\(\nabla \) :

Gradient operator

\(\text {Hess}\) :

Hessian operator

\({\mathcal {P}}\) :

Density manifold

\(\rho \) :

Probability density

\(\mu \) :

Reference density

\(T_\rho {\mathcal {P}}\) :

Tangent space

\(T_\rho ^*{\mathcal {P}}\) :

Cotangent space

\(g_\rho \) :

Density manifold metric tensor

\(\Delta _h=\nabla \cdot (h\nabla )\) :

Weighted Laplacian operator

\(\delta \) :

First \(L^2\) variation

\(\delta ^2\) :

Second \(L^2\) variation

\(\text {grad}_g\) :

Gradient operator

\({\text {Hess}}_g\) :

Hessian operator

\(\Gamma _\rho (\cdot , \cdot )\) :

Christoffel symbol

\((\rho , \sigma )\in T{\mathcal {P}}\) :

Tangent bundle

\((\rho , \Phi )\in T^*{\mathcal {P}}\) :

Cotangent bundle

\({\mathcal {D}}_\gamma \) :

\(\gamma \)-divergence

\({\mathcal {I}}_\gamma \) :

\(\gamma \)-Fisher information

\({\mathcal {W}}_\gamma \) :

\(\gamma \)-Wasserstein distance

\(L_\gamma \), \(L_\gamma ^*\) :

\(\gamma \)-Diffusion process generator

\(\Gamma _{\gamma , 1}\) :

\(\gamma \)-Gamma one operator

\(\Gamma _{\gamma , 2}\) :

\(\gamma \)-Gamma two operator

\(\kappa \) :

Log Sobolev constant

\(\lambda \) :

Poincare constant

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Funding

W. Li thanks the support of AFOSR MURI FA9550-18-1-0502, AFOSR YIP award No. FA9550-23-1-0087, NSF FRG grant: DMS-2245097, and NSF RTG grant: 2038080.

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  1. University of South Carolina, Columbia, SC, United States

    Wuchen Li

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Communicated by Jun Zhang.

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Li, W. Diffusion hypercontractivity via generalized density manifold. Info. Geo. (2023). https://doi.org/10.1007/s41884-023-00124-x

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