Title | Benamou–Brenier and duality formulas for the entropic cost on RCD*(K,N) spaces |

Publication Type | Journal Article |

Year of Publication | 2019 |

Authors | Gigli, N, Tamanini, L |

Journal | Probability Theory and Related Fields |

Date Published | Apr |

ISSN | 1432-2064 |

Abstract | In this paper we prove that, within the framework of $\textsf{RCD}^\star(K,N)$ spaces with $N<\infty$, the entropic cost (i.e. the minimal value of the Schrödinger problem) admits:A threefold dynamical variational representation, in the spirit of the Benamou–Brenier formula for the Wasserstein distance; A Hamilton–Jacobi–Bellman dual representation, in line with Bobkov–Gentil–Ledoux and Otto–Villani results on the duality between Hamilton–Jacobi and continuity equation for optimal transport;A Kantorovich-type duality formula, where the Hopf–Lax semigroup is replaced by a suitable `entropic' counterpart.We thus provide a complete and unifying picture of the equivalent variational representations of the Schrödinger problem as well as a perfect parallelism with the analogous formulas for the Wasserstein distance. Riemannian manifolds with Ricci curvature bounded from below are a relevant class of $\textsf{RCD}^*(K,N)$ spaces and our results are new even in this setting. |

URL | https://doi.org/10.1007/s00440-019-00909-1 |

DOI | 10.1007/s00440-019-00909-1 |

## Benamou–Brenier and duality formulas for the entropic cost on RCD*(K,N) spaces

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