Optimal transport

Classical optimal transport (Monge–Kantorovich) asks for the cheapest way to move a probability measure \(\mu\) onto a probability measure \(\nu\). Mass is conserved: every grain that leaves a source point must arrive somewhere. Behind a great deal of modern probability and PDE — the gradient-flow structure of the heat equation, hydrodynamic limits, displacement convexity — sits this conservation law.

Many real problems do not respect it. Re-balancing a portfolio is the canonical example: when capital is moved between assets the amount that arrives differs from the amount that left, because of fees, slippage, and tax. With Gabriela Kováčová and Niket Patel we set up a non-conservative optimal transport framework in which the transported mass is scaled by a factor depending on source and destination. We prove existence of optimal plans, strong duality, and existence of optimal maps in two regimes — perturbative mass change and quadratic mass loss — and derive a Benamou–Brenier-type dynamic formulation for \(\ell^p\) costs.

I am interested in extending the non-conservative theory to multi-marginal and entropic settings, in numerical schemes for the dynamic formulation, and in connections with the Otto calculus that drives much of the work on hydrodynamic limits and functional inequalities elsewhere on this page.

Further reading.

• Non-conservative optimal transport, with G. Kováčová, N. Patel. arXiv:2510.03332.