Long-time Behaviour of Entropic Interpolations

Abstract

In this article we investigate entropic interpolations. These measure valued curves describe the optimal solutions of the Schrödinger problem [Sch31], which is the problem of finding the most likely evolution of a system of independent Brownian particles conditionally to observations. It is well known that in the short time limit entropic interpolations converge to the McCann-geodesics of optimal transport. Here we focus on the long-time behaviour, proving in particular asymptotic results for the entropic cost and establishing the convergence of entropic interpolations towards the heat equation, which is the gradient flow of the entropy according to the Otto calculus interpretation. Explicit rates are also given assuming the Bakry-Émery curvature-dimension condition. In this respect, one of the main novelties of our work is that we are able to control the long time behavior of entropic interpolations assuming the CD(0, n) condition only.

Regularity properties of the Schrödinger cost

“…Using this lemma and the Theorem 3.1 we can easily extend the estimates provided in Theorem 1.4 of [9] and Theorem 3.6 of [7]. Note that in [9] the author has already extended the estimate which holds under the CD(ρ, ∞) curvature-dimension condition to the non-compact case, but we believe this is a pertinent example to illustrate the utility of Proposition 3.1.…”

“…In Theorem 3.6 of [7] and Theorem 1.4 of [9], estimates are provided for high values of T , but only in the case where both measures are compactly supported and smooth. Using the Proposition 3.1 we are able to extend these estimates to the non-compactly supported and non-smooth case.…”

“…These estimates are very useful, for instance they are fundamental to show the longtime convergence of entropic interpolations, see [7].…”

On the variational interpretation of local logarithmic Sobolev inequalities

“…Estimates under the (0, n)-convexity with n > 0, cf. [CCG20]. In this case we consider the map Φ(t) = λ(t) − tE T (x, y) and observe that (0, n) and Newton's law combined yield…”

“…Estimates under the (0, n)-convexity with n > 0, cf. [CCG20]. In this case we have that if (µ T t ) t∈[0,T ] is the entropic interpolation between µ and ν:…”