# Diffusion Hypercontractivity via Generalized Density Manifold

## TL;DR

This paper introduces a new framework connecting diffusion hypercontractivity with various inequalities through a generalized density manifold, employing advanced calculus and transport metrics.

## Contribution

It develops a one-parameter family of diffusion hypercontractivity results and introduces a novel PH^{-1}I inequality linking divergence, transport, and information measures.

## Key findings

- Established a new family of diffusion hypercontractivity inequalities.
- Derived the PH^{-1}I inequality connecting divergence, transport, and Fisher information.
- Presented a mean-field Bakry-Emery calculus and Yano's volume measure formula.

## Abstract

We prove a one-parameter family of diffusion hypercontractivity and present the associated Log-Sobolev, Poincare and Talagrand inequalities. A mean-field type Bakry-Emery iterative calculus and volume measure based integration formula (Yano's formula) are presented. Our results are based on the interpolation among divergence functional, generalized diffusion process, and generalized optimal transport metric. As a result, an inequality among Pearson divergence (P), negative Sobolev metric H^-1 and generalized Fisher information functional (I), named PH^{-1}I inequality, is derived.