A convergence rate for the entropic JKO scheme

The so-called JKO scheme, named after Jordan, Kinderlehrer and Otto, provides a variational way to construct discrete time approximations of certain partial differential equations (PDEs) appearing as gradient flows in the space of probability measures equipped with the Wasserstein metric. The method consists of an implicit Euler scheme, which can be implemented numerically. Yet, in practice, evaluating the Wasserstein distance can be numerically expensive. To address this problem, a common strategy introduced by Peyr\'e in 2015 and which has been shown to produce faster computations, is to replace the Wasserstein distance with its entropic regularization, also known as the Schr\"odinger cost. In 2026, the first author, Hraivoronska and Santambrogio, proved that if the regularization parameter $\varepsilon$ is proportional to the time step $\tau$, that is, $\varepsilon = \alpha \tau$ for some $\alpha > 0$, then as $\tau \to 0$, this change results in adding to the limiting PDE the additional linear diffusion term $\frac{\alpha}{2} \Delta \rho$. Our goal in this article is to provide a convergence rate under convexity assumptions between the entropic JKO scheme and the solution of the initial PDE as both $\alpha$ and $\tau$ tend to zero. This will appear as a consequence of a new bound between the classical and entropic JKO schemes.