The optimal transport problem, in its Benamou-Brenier representation, has a dual formulation involving the quadratic Hamilton-Jacobi equation. Reversing this perspective allows one to recover the optimal transport problem from the Hamilton-Jacobi equation via duality. A similar procedure can be applied to many other PDEs. Remarkably, except for certain degenerate cases (including classical OT), the transported probability measure is automatically replaced by a (non-negative-definite) matrix-valued measure. In 2018, Brenier demonstrated how this approach can be used to construct solutions to Cauchy problems for PDEs. We will discuss recent developments in this area and its links to Dafermos’ selection principle.
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