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

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).


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.)
Full Text: DOI EuDML