Optimal Ballistic Transport and Hopf-Lax Formulae on Wasserstein Space

Abstract

We investigate the optimal mass transport problem associated to the following "ballistic" cost functional on phase space $M\times M^{\ast }$, $$b_T(v,x):=\inf \{⟨v,\gamma (0)⟩+{\int }_0^{T}L(\gamma (t),\dot{\gamma }(t))dt,\gamma \in C^1([0,T),M),\gamma (T)=x\},$$ where $M={\mathbb{R}}^d$, $T>0$, and $L:M\times M\to \mathbb{R}$ 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 $M\times M$ by $$c_T(x,y):=\inf \{{\int }_0^{T}L(\gamma (t),\dot{\gamma }(t))dt,\gamma \in C^1([0,T),M),\gamma (0)=x,\gamma (T)=y\}.$$ We also point to links with the theory of mean field games.

arXiv comment25 pages, Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/

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