Ricci Curvature and Bochner Formulas for Martingales

Robert Haslhofer, Aaron Naber

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)1074-1108
Number of pages35
JournalCommunications on Pure and Applied Mathematics
Volume71
Issue number6
DOIs
StatePublished - Jun 2018

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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