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Dynamic and stochastic propagation of Brenier’s optimal mass transport

University of British Columbia

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.

Text

2018-04-25

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/65633

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/