A Concrete Estimate For The Weak Poincare Inequality On Loop Space

TL;DR

This paper provides a concrete estimate for the weak Poincare inequality on the loop space over a simply connected compact Riemannian manifold with positive Ricci curvature, using Malliavin calculus and Ornstein-Uhlenbeck operators.

Contribution

It offers a new explicit estimate for the weak Poincare inequality on loop spaces, advancing understanding of functional inequalities in infinite-dimensional geometric analysis.

Findings

Concrete estimate for weak Poincare inequality established

Results applicable to manifolds with positive Ricci curvature

Enhances analysis of Ornstein-Uhlenbeck operators on loop space

Abstract

The aim of the paper is to study the pinned Wiener measure on the loop space over a simply connected compact Riemannian manifold together with the Hilbert space structure induced by Mallianvin calculus and the induced Ornstein- Uhlenbeck operator. We give a concrete estimate for the weak Poincare inequality for the O-U Dirichlet form on loop space over simply connected compact Riemannian manifold with strict positive Ricci curvature.