The intrinsic dynamics of optimal transport.
(La dynamique intrinsèque du transport optimal.)

*(English. French summary)*Zbl 1364.49060The main objective of the authors in this paper can be formulated as follows: characterize cost functions on compact Riemannian manifolds for which uniqueness of the optimal plan occurs in the Monge-Kantorovich optimal transport problem.

Monge’s problem in the theory of optimal transport (remaining in the setting of the present paper) can be formulated as follows. Given \(M,N\) compact Riemannian manifolds, \(\mu,\nu\) probability measures on \(M\) and \(N\) respectively and a continuous cost function \(c:M\times N\to\mathbb{R}\), find an optimizer for the problem \[ \inf_{T_\#\mu=\nu}\int_M c(x,T(x))d\mu(x), \] where \(T:M\to N\) is a measurable map and \(T_\#\mu=\nu\) means that \(T\) pushes forward \(\mu\) onto \(\mu\).

We know that the above problem does not have always a solution. A relaxation of it is called the Kantorovich problem and it reads as \[ \inf_{\gamma\in\Pi(\mu,\nu)}\int_{M\times N} c(x,y)d\gamma(x,y), \] where \(\Pi(\mu,\nu)\) denotes the set of all transport plans, i.e. probability measures on \(M\times N\) with marginals \(\mu\) and \(\nu\). In contrast to Monge’s problem, this one always has a solution. Moreover, if the optimal plan is concentrated on a graph, this will provide a solution for Monge’s problem.

The question of uniqueness of minimizers in the Kantorovich problem is highly non-trivial. A typical way to get uniqueness of the optimal plan (and existence and uniqueness of the optimal map in Monge’s problem) is to show that any optimal plan is concentrated on a graph. Sufficient conditions that guarantee this are: Lipschitz continuity of \(c\), absolute continuity of \(\mu\) w.r.t. the Lebesgue measure on \(M\) and a so-called TWIST condition.

As the first result of the present paper, the authors show that there are \(C^2\) cost functions satisfying a TWIST-type condition and absolutely continuous probability measures such that the optimal plan in the Kantorovich problem is unique but it is not concentrated on a graph.

Then, as the main result of the paper, they give a set of sufficient conditions on the cost functions, which ensure the uniqueness of the optimal plans, for a generic class of measures \(\mu\) and \(\nu\) and manifolds \(M\) and \(N\). In particular, these conditions are independent of the manifolds. Previous results from the literature of similar flavor required that at least one of the two manifolds is homeomorphic to a sphere. In this sense, the above described result represents a major improvement of the previous results.

Monge’s problem in the theory of optimal transport (remaining in the setting of the present paper) can be formulated as follows. Given \(M,N\) compact Riemannian manifolds, \(\mu,\nu\) probability measures on \(M\) and \(N\) respectively and a continuous cost function \(c:M\times N\to\mathbb{R}\), find an optimizer for the problem \[ \inf_{T_\#\mu=\nu}\int_M c(x,T(x))d\mu(x), \] where \(T:M\to N\) is a measurable map and \(T_\#\mu=\nu\) means that \(T\) pushes forward \(\mu\) onto \(\mu\).

We know that the above problem does not have always a solution. A relaxation of it is called the Kantorovich problem and it reads as \[ \inf_{\gamma\in\Pi(\mu,\nu)}\int_{M\times N} c(x,y)d\gamma(x,y), \] where \(\Pi(\mu,\nu)\) denotes the set of all transport plans, i.e. probability measures on \(M\times N\) with marginals \(\mu\) and \(\nu\). In contrast to Monge’s problem, this one always has a solution. Moreover, if the optimal plan is concentrated on a graph, this will provide a solution for Monge’s problem.

The question of uniqueness of minimizers in the Kantorovich problem is highly non-trivial. A typical way to get uniqueness of the optimal plan (and existence and uniqueness of the optimal map in Monge’s problem) is to show that any optimal plan is concentrated on a graph. Sufficient conditions that guarantee this are: Lipschitz continuity of \(c\), absolute continuity of \(\mu\) w.r.t. the Lebesgue measure on \(M\) and a so-called TWIST condition.

As the first result of the present paper, the authors show that there are \(C^2\) cost functions satisfying a TWIST-type condition and absolutely continuous probability measures such that the optimal plan in the Kantorovich problem is unique but it is not concentrated on a graph.

Then, as the main result of the paper, they give a set of sufficient conditions on the cost functions, which ensure the uniqueness of the optimal plans, for a generic class of measures \(\mu\) and \(\nu\) and manifolds \(M\) and \(N\). In particular, these conditions are independent of the manifolds. Previous results from the literature of similar flavor required that at least one of the two manifolds is homeomorphic to a sphere. In this sense, the above described result represents a major improvement of the previous results.

Reviewer: Alpár R. Mészáros (Los Angeles)

##### MSC:

49Q20 | Variational problems in a geometric measure-theoretic setting |

28A35 | Measures and integrals in product spaces |

49J45 | Methods involving semicontinuity and convergence; relaxation |

##### Keywords:

optimal transport; Monge-Kantorovitch problem; optimal transport map; optimal transport plan; numbered limb system; sufficient conditions for uniqueness##### References:

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