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Ricci curvature and Bochner formulas for martingales. (English) Zbl 1393.60042

In this paper, the authors generalize the classical Bochner formula for the heat flow on \(M\) to martingales on the path space \(PM\). Also, the authors explain how bounded Ricci curvature can be understood by analyzing the evolution of martingales on path space generalizing the well known and important principles of how lower bounds on Ricci curvature can be understood by analyzing the heat flow. One of the main results of the paper is: their 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, the authors obtain new characterizations of bounded Ricci curvature, new gradient estimates for martingales on path space, new Hessian estimates for martingales on path space, and also streamlined proofs of the previous characterizations of bounded Ricci curvature.

MSC:

60G44 Martingales with continuous parameter
53B20 Local Riemannian geometry
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