Open Collections

Abstract

I present analysis of how the mass transports that optimize the inner product cost---considered by Y. Brenier---propagate in time along a given Lagrangian in both deterministic and stochastic settings. While for the minimizing transports one may easily obtain Hopf-Lax formulas on Wasserstein space by inf-convolution, this is not the case for the maximizing transports, which are sup-inf problems. In this case, we assume that the Lagrangian is jointly convex on phase space, which allow us to use Bolza-type duality, a well known phenomenon in the deterministic case but, as far as I know, novel in the stochastic case. Hopf-Lax formulas help relate optimal ballistic transports to those associated with the dynamic fixed-end transports studied by Bernard-Buffoni and Fathi-Figalli in the deterministic case, and by Mikami-Thieullen in the stochastic setting.