Remarks on spectral gaps on the Riemannian path space

Shizan Fang,Bo Wu

doi:10.1214/17-ecp51

Abstract

In this paper, we will give some remarks on links between the spectral gap of the Ornstein-Uhlenbeck operator on the Riemannian path space with lower and upper bounds of the Ricci curvature on the base manifold; this work was motivated by a recent work of A. Naber on the characterization of the bound of the Ricci curvature by analysis of path spaces.

Highlights

Let M be a complete smooth Riemannian manifold of dimension d, and Z a C1-vector field on M
We will be concerned with the diffusion operator
The purpose in [21] is to proceed in the opposite direction, to get the bound for Ricci curvature tensor of the base manifold M from the analysis of the path space WxT (M )

Summary

Introduction

Let M be a complete smooth Riemannian manifold of dimension d, and Z a C1-vector field on M. Naber proved that if the uniform bound (1.6) holds, the Ricci curvature of the base manifold has an upper bound It is well-known that Inequality (1.2) implies the lower bound (1.1), Condition (1.6) implies (1.5). The logarithmic Sobolev inequality for Dτ F defined in (1.9) has been established in [2], as well as in [20] or [6] where the constant was estimated using the bound of Ricci curvature tensor of the base manifold M. The purpose in [21] is to proceed in the opposite direction, to get the bound for Ricci curvature tensor of the base manifold M from the analysis of the path space WxT (M ). + o(T ), as T → 0 under the following condition (3.1)

Framework of the Riemannian path space

Precise lower bound on the spectral gap