## Abstract

We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the L^{2}-space produces the same evolution as the gradient flow of the relative entropy in the L^{2}-Wasserstein space. This means that the heat flow is well-defined by either one of the two gradient flows. Combining properties of these flows, we are able to deduce the Lipschitz continuity of the heat kernel as well as Bakry-Émery gradient estimates and the Γ_{2}-condition. Our identification is established by purely metric means, unlike preceding results relying on PDE techniques. Our approach generalizes to the case of heat flow with drift.

Original language | English |
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Pages (from-to) | 307-331 |

Number of pages | 25 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 66 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2013 Mar |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics