A formula for the time derivative of the entropic cost and applications

TL;DR

This paper investigates the regularity and derivatives of the entropic cost in the Schrödinger problem, providing explicit formulas and applications to large-time limits, convergence rates, and improved Taylor expansions.

Contribution

It offers a detailed analysis of the entropic cost’s derivatives with respect to time, including explicit formulas and applications to classical and mean field Schrödinger problems.

Findings

Explicit formulas for first and second derivatives of entropic cost

Large-time limit and exponential convergence rates for entropic cost

Improved second-order Taylor expansion of T→T𝓒_T around T=0

Abstract

In the recent years the Schr\"odinger problem has gained a lot of attention because of the connection, in the small-noise regime, with the Monge-Kantorovich optimal transport problem. Its optimal value, the \emph{entropic cost} $\mathscr{C}_T$, is here deeply investigated. In this paper we study the regularity of $\mathscr{C}_T$ with respect to the parameter $T$ under a curvature condition and explicitly compute its first and second derivative. As applications: - we determine the large-time limit of $\mathscr{C}_T$ and provide sharp exponential convergence rates; we obtain this result not only for the classical Schr\"odinger problem but also for the recently introduced Mean Field Schr\"odinger problem [3]; - we improve the Taylor expansion of $T \mapsto T\mathscr{C}_T$ around $T=0$ from the first to the second order.