In this paper we prove that, within the framework of RCD∗(K, N) spaces with N< ∞, 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 RCD∗(K, N) spaces and our results are new even in this setting.

Benamou–Brenier and duality formulas for the entropic cost on RCD∗(K, N) spaces / Gigli, N.; Tamanini, L.. - In: PROBABILITY THEORY AND RELATED FIELDS. - ISSN 0178-8051. - 176:1-2(2020), pp. 1-34. [10.1007/s00440-019-00909-1]

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

Gigli N.;Tamanini L.
2020

Abstract

In this paper we prove that, within the framework of RCD∗(K, N) spaces with N< ∞, 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 RCD∗(K, N) spaces and our results are new even in this setting.
176
1-2
1
34
https://link.springer.com/article/10.1007/s00440-019-00909-1
https://arxiv.org/abs/1805.06325v1
Gigli, N.; Tamanini, L.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/20.500.11767/111348
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