2018/04/24 by Barton, Alistair, Ghoussoub, Nassif
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1804.10052
We investigate how mass transports that optimize the inner product cost -considered by Y. Brenier- propagate in time along a given Lagrangian. In the deterministic case, we consider transports that maximize and minimize the following "ballistic" cost functional on phase space M^*× M, bT(v, x):=inf\⟨ v, γ(0)⟩ +∫0TL(t, 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 suitable Lagrangian. We also consider the stochastic counterpart: % \underlineBTs(μ,ν):=inf\𝔼[⟨ V,X0⟩ +∫0T L(t, X,β(t,X)) dt]; X∈ A, V∼μ,XT∼ ν\ where A is the set of stochastic processes satisfying dX=βX(t,X) dt+ dWt, for some drift βX(t,X), and where Wt is σ(Xs:0≤ s≤ t)-Brownian motion. While inf-convolution allows us to easily obtain Hopf-Lax formulas on Wasserstein space for cost minimizing transports, this is not the case for total cost maximizing transports, which actually are sup-inf problems. However, in the case where the Lagrangian L is jointly convex on phase space, Bolza-type dualities --well known in the deterministic case but novel in the stochastic case--transform sup-inf problems to sup-sup settings. Hopf-Lax formulas relate optimal ballistic transports to those associated with dynamic fixed-end transports studied by Bernard-Buffoni and Fathi-Figalli in the deterministic case, and by Mikami-Thieullen in the stochastic setting. We also write Eulerian formulations and point to links with the theory of mean field games.