Functional inequalities on path space over a non-compact Riemannian manifold

TL;DR

This paper establishes functional inequalities and constructs Dirichlet forms on path space over non-compact Riemannian manifolds, extending analysis tools to settings with unbounded Ricci curvature and diffusion coefficients.

Contribution

It introduces new Dirichlet forms and proves key inequalities on path space over non-compact manifolds with unbounded curvature.

Findings

Existence of O-U and damped O-U Dirichlet forms on path space.

Weighted log-Sobolev and Poincaré inequalities established.

Construction of quasi-regular local Dirichlet forms with unbounded coefficients.

Abstract

We prove the existence of the O-U Dirichlet form and the damped O-U Dirichlet form on path space over a general non-compact Riemannian manifold which is complete and stochastically complete. We show a weighted log-Sobolev inequality for the O-U Dirichlet form and the (standard) log-Sobolev inequality for the damped O-U Dirichlet form. In particular, the Poincar\'e inequality (and the super Poincar\'e inequality) can be established for the O-U Dirichlet form on path space over a class of Riemannian manifolds with unbounded Ricci curvatures. Moreover, we construct a large class of quasi-regular local Dirichlet forms with unbounded random diffusion coefficients on the path space over a general non-compact manifold.