In this paper we prove that, within the framework of RCD*(K, N) spaces with N < infinity, the entropic cost (i.e. the minimal value of the Schrodinger 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 Schrodinger 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 RCD* (K, N) spaces and our results are new even in this setting.

Gigli, N., Tamanini, L., Benamou–Brenier and duality formulas for the entropic cost on RCD∗(K, N) spaces, <<PROBABILITY THEORY AND RELATED FIELDS>>, 2020; 176 (1-2): 1-34. [doi:10.1007/s00440-019-00909-1] [https://hdl.handle.net/10807/232497]

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

Tamanini, Luca
2020

Abstract

In this paper we prove that, within the framework of RCD*(K, N) spaces with N < infinity, the entropic cost (i.e. the minimal value of the Schrodinger 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 Schrodinger 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 RCD* (K, N) spaces and our results are new even in this setting.
2020
Inglese
Gigli, N., Tamanini, L., Benamou–Brenier and duality formulas for the entropic cost on RCD∗(K, N) spaces, <<PROBABILITY THEORY AND RELATED FIELDS>>, 2020; 176 (1-2): 1-34. [doi:10.1007/s00440-019-00909-1] [https://hdl.handle.net/10807/232497]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10807/232497
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