A generalized Dacorogna-Moser construction and the problem of geodesics in the space of couplings
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112
Braconnier
The seminal work by Dacorogna and Moser introduced a way to
construct transport maps with high regularity from a probability
distribution to another on a bounded domain. In this work, we
generalize the Dacorogna-Moser construction to strictly
asymptotically log-concave measures on non-compact domains, by
establishing uniform-in-time estimates on parabolic and elliptic
PDEs defined on the whole space. Then, we study the application
of this result to the analysis of the problem of geodesics in the
space of couplings with two given marginals, solving an open
question in Conforti, Lacker, Pal [JEMS'25]. In particular we study
the existence of minimizers, the existence of a Lagrange multiplier
associated to the marginal constraints and the optimality conditions;
furthermore, we introduce a suitable regularization and we prove Γ-
convergence to the original problem and weak-* convergence of the
Lagrange multipliers in a proper space, similarly to what is done in
Baradat, Monsaingeon [ARMA'20]