2017/05/16 by Nassif Ghoussoub, Ghoussoub, Nassif
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1705.05951
openalex publication_date 2017/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the optimal mass transport problem associated to the following "ballistic" cost functional on phase space M× M^*, bT(v, x):=inf\⟨ v, γ(0)⟩ +∫0TL(γ(t), γ(t)) dt, γ∈ C1([0, T), M), γ(T)=x\, where M=ℝd, T>0, and L:M× M → ℝ is a Lagrangian that is jointly convex in both variables. Under suitable conditions on the initial and final probability measures, we use convex duality à la Bolza and Monge-Kantorovich theory to lift classical Hopf-Lax formulae from state space to Wasserstein space. This allows us to relate optimal transport maps for the ballistic cost to those associated with the fixed-end cost defined on M× M by cT(x,y):=inf\∫0TL(γ(t), γ(t)) dt, γ∈ C1([0, T), M), γ(0)=x, γ(T)=y\. We also point to links with the theory of mean field games.