Recently, the vanishing-step-size limit of the Sinkhorn algorithm at finite regularization parameter $\varepsilon$ was shown to be a mirror descent in the space of probability measures. We give $L^2$ contraction criteria in two time-dependent metrics induced by the mirror Hessian, which reduce to the coercivity of certain conditional expectation operators. We then give an exact identity for the entropy production rate of the Sinkhorn flow, which was previously known only to be nonpositive. Examining this rate shows that the standard semigroup analysis of diffusion processes extends systematically to the Sinkhorn flow. We show that the flow induces a reversible Markov dynamics on the target marginal as an Onsager gradient flow. We define the Dirichlet form associated to its (nonlocal) infinitesimal generator, prove a Poincaré inequality for it, and show that the spectral gap is strictly positive along the Sinkhorn flow whenever $\varepsilon > 0$. Lastly, we show that the entropy decay is exponential if and only if a logarithmic Sobolev inequality (LSI) holds. We give for illustration two immediate practical use-cases for the Sinkhorn LSI: as a design principle for the latent space in which generative models are trained, and as a stopping heuristic for discrete-time algorithms.

We prove that the sequence of marginals obtained from the iterations of the Sinkhorn algorithm or the iterative proportional fitting procedure (IPFP) on joint densities, converges to an absolutely continuous curve on the $2$-Wasserstein space, as the regularization parameter $\varepsilon$ goes to zero and the number of iterations is scaled as $1/\varepsilon$ (and other technical assumptions). This limit, which we call the Sinkhorn flow, is an example of a Wasserstein mirror gradient flow, a concept we introduce here inspired by the well-known Euclidean mirror gradient flows. In the case of Sinkhorn, the gradient is that of the relative entropy functional with respect to one of the marginals and the mirror is half of the squared Wasserstein distance functional from the other marginal. Interestingly, the norm of the velocity field of this flow can be interpreted as the metric derivative with respect to the linearized optimal transport (LOT) distance. An equivalent description of this flow is provided by the parabolic Monge-Amp\`{e}re PDE whose connection to the Sinkhorn algorithm was noticed by Berman (2020). We derive conditions for exponential convergence for this limiting flow. We also construct a Mckean-Vlasov diffusion whose marginal distributions follow the Sinkhorn flow.