Hessian stability and convergence rates for entropic and Sinkhorn potentials via semiconcavity

In this paper we determine quantitative stability bounds fo r the Hessian of entropic potentials, i.e., the dual solution to the entropic optimal transport proble m. Up to authors' knowledge this is the first work addressing this second-orde r quantitative stability estimate in general unbounded settings. Our proof strategy relies on se miconcavity properties of entropic potentials and on the representation of entropic transport plans as laws of forward and backward diffusion processes, known as Schr odinger bridges. Moreov er, our approach allows to deduce a stochastic proof of quantitative stability entropic estim ates and integrated gradient estimates as well. Finally, as a direct consequence of these stability bounds, we deduce exponential convergence rates for gradient and Hessian of Sinkhorn iter ates along Sinkhorn's algorithm, a problem that was still open in unbounded settings. Our rates have a polynomial dependence on the regularization parameter.