Lévy-Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator. (English) Zbl 0855.58011

Bakry, D.; Ledoux, M.

Invent. Math. 123, No. 2, 259-281 (1996).

The authors establish a Lévy-Gromov isoperimetric inequality for the invariant measure of an infinite-dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature (Theorem 5.1).

Reviewer: N.Papaghiuc (Iaşi)

MSC:

58D20	Measures (Gaussian, cylindrical, etc.) on manifolds of maps
28C20	Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)

Keywords:

Lévy-Gromov isoperimetric inequality; invariant measure; infinite-dimensional diffusion generator; Gaussian measure; Gaussian isoperimetric inequality; Sobolev inequality; heat kernel measures; Wiener measure