Schrödinger's problem and its links with optimal transport and functional inequalities.

Summary During the last 20 years, optimal transport theory has proven to be an efficient tool to study the asymptotic behavior of diffusion equations, to prove functional inequalities and to extend geometric properties in extremely general spaces such as measured metric spaces, etc. The curvature-dimensional condition of the Bakry-Emery theory appears to be the cornerstone of these applications. The curvature-dimension condition of the Bakry-Emery theory appears as the cornerstone of these applications. It suffices to think of the simplest and most important case of the Wasserstein quadratic distance W2 : the contraction of the heat flow in W2 characterizes the uniform lower bounds for the Ricci curvature . the Talagrand inequality of the transport, comparing W2 to the relative entropy is implied and implies, by the HWI inequality, the log-Sobolev inequality . McCann geodesics in Wasserstein space (P2(Rn),W2) allow to prove important functional properties such as convexity, and standard functional inequalities such as isoperymmetry, measure concentration properties, Prekopa-Leindler inequality and so on. Nevertheless, the lack of regularity of the minimization schemes requires non-smooth analysis arguments. Schrödinger's problem is an entropy minimization problem with fixed marginal constraints and reference process. From large deviation theory, when the reference process is Brownian motion, its minimum value A converges to W2 when the temperature is zero. Entropic interpolations, solutions of the Schrödinger problem, are characterized in terms of Markov semigroups, which naturally implies Γ2 calculations and the curvature-dimension condition. Dating back to the 1930s and neglected for decades, the Schrodinger problem has been gaining popularity in recent years in various fields, thanks to its relation to optimal transport, the regularity of its solutions, and other powerful properties in numerical calculations. The aim of this work is twofold. First, we study some analogies between the Schrödinger problem and the optimal transport providing new proofs of the dual Kantorovich formulation and the dynamic Benamou-Brenier formulation for the entropy cost A. Then, as an application of these connections, we derive some functional properties and inequalities under curvature-dimension conditions. In particular, we prove the concavity of the exponential entropy along the entropy interpolations under the curvature-dimension condition CD(0, n) and the regularity of the entropy cost along the heat flow. We also give various proofs of the evolutionary variational inequality for A and the contraction of the heat flow in A, recovering as a limiting case, the classical results in W2 under CD(κ,∞) and CD(0, n). Finally, we propose a simple proof of the Gaussian concentration property via the Schrödinger problem as an alternative to classical arguments such as the Marton argument based on optimal transport.