Seminario - Uniform Estimates and Classical Solutions for Mean-Field Planning Problem

mercoledì 6 luglio alle 12:00 in Aula G il professore Bakaryan (KAUST) terrà un seminario dal titolo "Uniform Estimates and Classical Solutions for Mean-Field Planning Problem"

Abstract: We explore new estimates for the first-order mean-field planning (MFP) problems using displacement convexity. Displacement convexity is a fundamental tool in optimal transport that often reveals hidden convexity of functionals. Taking into account the similarities between the Benamou-Brenier formulation of optimal transport and MFP, we extend displacement convexity methods to MFP. In particular, we identify a class of functions, that depend on solutions of MFP, that are convex in time and, thus, obtain new a priori bounds for solutions of MFP. This convexity gives bounds for the density of solutions to the MFP problem with a potential and provides conditions that ensure lower bounds for the density.  Finally, relying on our estimates, we prove the existence and uniqueness of solutions for a particular first-order MFP problem with a potential.