Using Sinkhorn in the JKO Scheme Adds Linear Diffusion

The JKO (Jordan–Kinderlehrer–Otto) scheme is a time-discrete scheme of implicit Euler type that allows one to construct weak solutions of evolution PDEs which have a Wasserstein gradient structure. The purpose of this work is to study the effect of replacing the classical quadratic optimal transport problem by the Schrödinger problem (a.k.a. the entropic regularization of optimal transport, efficiently computed by the Sinkhorn algorithm) at each step of this scheme. We find that if [Formula: see text] is the regularization parameter of the Schrödinger problem, and [Formula: see text] is the time-step parameter, considering the limit [Formula: see text] with [Formula: see text] results in adding the term [Formula: see text] on the right-hand side of the limiting PDE. In the case [Formula: see text] we improve a previous result by Carlier et al. [ SIAM J. Math. Anal., 49 (2017), pp. 1385–1418].