Optimal transport consists in sending a given source probability measure ρ to a given target probability measure μ in an optimal way with respect to a certain cost. Optimal transport has been widely used in many fields, including analysis, probability, statistics, geometry, and optimization. Under classical assumptions, there exists a unique optimal transport map from ρ to μ (Brenier's, McCann's theorems, etc.). In this talk based on a collaboration with Quentin Mérigot, we provide a quantitative answer to the following stability question, notably motivated by numerical analysis and statistics: if μ is perturbed, can the optimal transport map from ρ to μ change significantly? The answer depends on the properties of the source measure ρ. We will also explain some mechanisms leading to instability and present a few conjectures.
NB: This talk is part of the activity of the MUR Excellence Department Project MATH@TOV CUP E83C23000330006