Ricci Curvature and Bochner Formulas for Martingales

TL;DR

This paper extends the classical Bochner formula to martingales on path space, linking Ricci curvature bounds to martingale behavior and providing new estimates and characterizations.

Contribution

It introduces a generalized Bochner formula for martingales on path space and connects Ricci curvature bounds to martingale estimates, offering new characterizations and streamlined proofs.

Findings

New gradient estimates for martingales on path space

Hessian estimates for martingales on path space

Characterizations of bounded Ricci curvature

Abstract

We generalize the classical Bochner formula for the heat flow on M to martingales on the path space PM, and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two sided bounds on Ricci curvature in much the same manner that the classical Bochner formula on M is related to lower bounds on Ricci curvature. Using this formalism, we obtain new characterizations of bounded Ricci curvature, new gradient estimates for martingales on path space, new Hessian estimates for martingales on path space, and streamlined proofs of the previous characterizations of bounded Ricci curvature of the second author (arXiv:1306.6512).