The Hamilton-Jacobi semigroup on length spaces and applications

2006/12/19 by John Lott, Cédric Villani, Lott, John +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #Differential Geometry (math.DG) #Diffusion and Search Dynamics #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.math/0612560

openalex publication_date 2006/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define a Hamilton-Jacobi semigroup acting on continuous functions on a compact length space. Following a strategy of Bobkov, Gentil and Ledoux, we use some basic properties of the semigroup to study geometric inequalities related to concentration of measure. Our main results are that (1) a Talagrand inequality on a measured length space implies a global Poincare inequality and (2) if the space satisfies a doubling condition, a local Poincare inequality and a log Sobolev inequality then it also satisfies a Talagrand inequality.

Citations

• Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

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