@article {Gigli2019,
	title = {Benamou{\textendash}Brenier and duality formulas for the entropic cost on RCD*(K,N) spaces},
	journal = {Probability Theory and Related Fields},
	year = {2019},
	month = {Apr},
	abstract = {<p>In this paper we prove that, within the framework of $\textsf{RCD}^\star(K,N)$ spaces with $N\&lt;\infty$, the entropic cost (i.e. the minimal value of the Schr{\"o}dinger problem) admits:A threefold dynamical variational representation, in the spirit of the Benamou{\textendash}Brenier formula for the Wasserstein distance; A Hamilton{\textendash}Jacobi{\textendash}Bellman dual representation, in line with Bobkov{\textendash}Gentil{\textendash}Ledoux and Otto{\textendash}Villani results on the duality between Hamilton{\textendash}Jacobi and continuity equation for optimal transport;A Kantorovich-type duality formula, where the Hopf{\textendash}Lax semigroup is replaced by a suitable {\textquoteleft}entropic{\textquoteright} counterpart.We thus provide a complete and unifying picture of the equivalent variational representations of the Schr{\"o}dinger 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.</p>},
	issn = {1432-2064},
	doi = {10.1007/s00440-019-00909-1},
	url = {https://doi.org/10.1007/s00440-019-00909-1},
	author = {Nicola Gigli and Luca Tamanini}
}
@article {GENTIL2019,
	title = {An entropic interpolation proof of the HWI inequality},
	journal = {Stochastic Processes and their Applications},
	year = {2019},
	abstract = {<p>The HWI inequality is an {\textquotedblleft}interpolation{\textquotedblright}inequality between the Entropy H, the Fisher information I and the Wasserstein distance W. We present a pathwise proof of the HWI inequality which is obtained through a zero noise limit of the Schr{\"o}dinger problem. Our approach consists in making rigorous the Otto{\textendash}Villani heuristics in Otto and Villani (2000) taking advantage of the entropic interpolations, which are regular both in space and time, rather than the displacement ones.</p>},
	keywords = {Entropic interpolations, Fisher information, Relative entropy, Schr{\"o}dinger problem, Wasserstein distance},
	issn = {0304-4149},
	doi = {https://doi.org/10.1016/j.spa.2019.04.002},
	url = {http://www.sciencedirect.com/science/article/pii/S0304414918303454},
	author = {Ivan Gentil and Christian L{\'e}onard and Luigia Ripani and Luca Tamanini}
}
@article {gigli2018second,
	title = {Second order differentiation formula on RCD(K, N) spaces},
	journal = {Rendiconti Lincei-Matematica e Applicazioni},
	volume = {29},
	number = {2},
	year = {2018},
	pages = {377{\textendash}386},
	author = {Nicola Gigli and Luca Tamanini}
}
@article {1802.02463,
	title = {Second order differentiation formula on RCD*(K,N) spaces},
	year = {2018},
	author = {Nicola Gigli and Luca Tamanini}
}
@article {1701.03932,
	title = {Second order differentiation formula on compact RCD*(K,N) spaces},
	year = {2017},
	author = {Nicola Gigli and Luca Tamanini}
}
