Uniform large-scale $\varepsilon$-regularity for entropic optimal transport

We study the regularity properties of the minimisers of entropic optimal transport providing a natural analogue of the $\varepsilon$-regularity theory of quadratic optimal transport in the entropic setting. More precisely, we show that if the minimiser of the entropic problem satisfies a gradient BMO-type estimate at some scale, the same estimate holds all the way down to the natural length-scale associated to the entropic regularisation. Our result follows from a more general $\varepsilon$-regularity theory for optimal transport costs which can be viewed as perturbations of quadratic optimal transport. We consider such a perturbed cost and require that, under a certain class of admissible affine rescalings, the minimiser remains a local quasi-minimiser of the quadratic problem (in an appropriate sense) and that the cost of "long trajectories" of minimisers (and their rescalings) is small. Under these assumptions, we show that the minimiser satisfies an appropriate $C^{2,\alpha}$ Morrey$\unicode{x2013}$Campanato-type estimate which is valid up to the scale of quasi-minimality.