Entropic regularization of continuous optimal transport problems

We analyze continuous optimal transport problems in the so-called Kantorovich form, where we seek a plan between two marginals that are probability measures on compact subsets of Euclidean space. consider case regularization with negative entropy respect to Lebesgue measure, which has attracted attention because it can be solved by very simple Sinkhorn algorithm. first regularized problem context classical Fenchel duality and derive strong result for predual space functions. However, this may not admit minimizer, prevents obtaining primal-dual optimality conditions. then show primal is naturally analyzed Orlicz functions finite sense entropically admits minimizer if only have entropy. dual corresponding space, existence shown purely variational arguments conditions derived. For do entropy, finally Gamma-convergence smoothed original problem.