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arXiv · 2402.04159

Optimal transport in the frame of abstract Lax-Oleinik operator revisited

Wei Cheng·Jiahui Hong·Tianqi Shi
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Abstract

This is our first paper on the extension of our recent work on the Lax-Oleinik commutators and its applications to the intrinsic approach of propagation of singularities of the viscosity solutions of Hamilton-Jacobi equations. We reformulate Kantorovich-Rubinstein duality theorem in the theory of optimal transport in terms of abstract Lax-Oleinik operators, and analyze the relevant optimal transport problem in the case the cost function $c(x,y)=h(t_1,t_2,x,y)$ is the fundamental solution of Hamilton-Jacobi equation. For further applications to the problem of cut locus and propagation of singularities in optimal transport, we introduce corresponding random Lax-Oleinik operators. We also study the problem of singularities for $c$-concave functions and its dynamical implication when $c$ is the fundamental solution with $t_2-t_1\ll1$ and $t_2-t_1<\infty$, and $c$ is the Peierls' barrier respectively.

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Wei Cheng, Jiahui Hong, Tianqi Shi. 2024-02-06. Optimal transport in the frame of abstract Lax-Oleinik operator revisited. https://arxiv.org/abs/2402.04159

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