Let Wo(M) be the space of paths of unit time length on a connected, complete Riemannian manifold M such that γ(0) = o, a fixed point on M, and v the Wiener measure on Wo(M) (the law of Brownian motion on M starting at o). If the Ricci curvature is bounded by c, then the following logarithmic Sobolev inequality holds: ∫Wo(M) F2 log |F|dv ≤ e3c∥DF∥2 + ∥F∥2 log ∥F∥.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics