Optimal Ballistic Transport and Hopf-Lax Formulae on Wasserstein Space

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 `a 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$$ c_T(x,y):=\inf{\int_0^TL(\gamma(t), {\dot \gamma}(t)), dt, \gamma\in C^1([0, T), M), \gamma(0)=x, \gamma(T)=y}. $$ We also point to links with the theory of mean field games.