Coupling of Brownian motions and Perelman's L-functional

Kazumasa Kuwada, Robert Philipowski

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that their normalized L-distance is a supermartingale. As a corollary, we obtain the monotonicity of the transportation cost between two solutions of the heat equation in the case that the cost function is the composition of a concave non-decreasing function and the normalized L-distance. In particular, it provides a new proof of a recent result of Topping [P. Topping, L-optimal transportation for Ricci flow, J. Reine Angew. Math. 636 (2009) 93-122].

Original languageEnglish
Pages (from-to)2742-2766
Number of pages25
JournalJournal of Functional Analysis
Volume260
Issue number9
DOIs
Publication statusPublished - 2011 May 1

Keywords

  • Brownian motion
  • Coupling
  • L-functional
  • Ricci flow

ASJC Scopus subject areas

  • Analysis

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