Spectral gap on Riemannian path space over static and evolving manifolds

TL;DR

This paper estimates the spectral gap of the Ornstein-Uhlenbeck operator on path space over Riemannian manifolds with pinched Ricci curvature, including evolving manifolds, extending previous work and analyzing short-time asymptotics.

Contribution

It provides explicit estimates and short-time asymptotics for the spectral gap on path space over static and evolving Riemannian manifolds, extending prior results.

Findings

Explicit spectral gap estimates for static manifolds

Analysis of short-time asymptotics of the spectral gap

Extension of results to manifolds under geometric flow

Abstract

In this article, we continue the discussion of Fang-Wu (2015) to estimate the spectral gap of the Ornstein-Uhlenbeck operator on path space over a Riemannian manifold of pinched Ricci curvature. Along with explicit estimates we study the short-time asymptotics of the spectral gap. The results are then extended to the path space of Riemannian manifolds evolving under a geometric flow. Our paper is strongly motivated by Naber's recent work (2015) on characterizing bounded Ricci curvature through stochastic analysis on path space.