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We develop a mathematical theory of entropic regularisation of unbalancedoptimal transport problems. Focusing on static formulation and relying on theformalism developed for the unregularised case, we show that unbalanced optimaltransport problems can be regularised in two qualitatively distinct ways -either on the original space or on the extended space. We derive severalreformulations of the two regularised problems and in particular introduce theidea of a regularised induced marginal perspective cost function allowing us toderive an extended space formulation of the original space regularisation. Wealso prove convergence to the unregularised problem in the case of the extendedspace regularisation and discuss on-going work on deriving a unified frameworkbased on higher order liftings in which both regularisations can be directlycompared. We also briefly touch upon how these concepts translate to thecorresponding dynamic formulations and provide evidence why the extended spaceregularisation should be preferred. This is a preliminary version of themanuscript, to be updated in the near future.

Entropic regularisation of unbalanced optimal transportation problems