# Ricci Curvature and Infinite Dimensional Analysis on Path Space

# Ricci Curvature and Infinite Dimensional Analysis on Path Space

In this talk we discuss recent connections between the Ricci curvature of a Riemannian manifold and the analysis on the path space of the manifold. We will see that bounded Ricci curvature controls the analysis on the path space P(M) of a manifold in much the same way that lower Ricci curvature controls the analysis on M itself. In fact, the estimates are not only implied by bounded Ricci curvature, but turn out to be equivalent to bounded Ricci curvature, and thus characterize bounds on the Ricci curvature. There are three distinct such characterizations given. The first is a gradient estimate that behaves as an infinite dimensional analogue of the Bakry-Emery gradient estimate on path space. The second is a C^{1/2}-Holder estimate on the time regularity of the martingale decomposition of functions on path space. For the third we consider the Ornstein-Uhlenbeck operator, a form of infinite dimensional laplace operator, and show that bounded Ricci curvature is equivalent to an appropriate spectral gap on this operator. In the second part of the paper, which we will only briefly discuss here, we use this to make sense of bounded Ricci curvature on arbitrary metric-measure spaces, and prove that they have many of the same properties as smooth spaces. In particular, it is proved such spaces have a lower Ricci curvature bound in the sense of Lott-Villani-Sturm.