First and second order fluctutations in sparse optimal transport

C. A. Cardoso Correia Perello, A. González Sanz, M. Nutz

We determine the leading-order behaviour of the difference between sparse $L^p$-regularised optimal transport and classical quadratic optimal transport for Hölder-continuous source and target densities. Using the connection with the porous medium equation, we construct matching transport potentials and couplings that capture the asymptotic behaviour of the cost, which in turn yields the exact magnitude of the leading-order term. We further identify the order of the second-order term and show that it depends explicitly on the Hölder regularity of the measures, suggesting a natural, power-law ansatz for the small-time behaviour of the porous medium equation with Hölder-continuous initial data. Joint work with Alberto González Sanz and Marcel Nutz.

Keywords: Optimal transport, sparse optimal transport, porous medium equation

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