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Fathi–Mikulincer–Shenfeld conjecture on Caffarelli estimates under Lipschitz perturbations

For every L≥0L\ge 0, there exists a finite constant C(L)C(L), independent of the dimension dd, such that for every d≥1d\ge 1 and every globally LL-Lipschitz function B:Rd→RB:\mathbb{R}^d\to\mathbb{R}, the Brenier map TT transporting the standard Gaussian measure γd\gamma_d to νB:=ZB−1e−Bγd\nu_B:=Z_B^{-1}e^{-B}\gamma_d, where ZB:=∫Rde−B dγdZ_B:=\int_{\mathbb{R}^d}e^{-B}\,d\gamma_d, admits a globally Lipschitz representative satisfying Lip⁡(T)≤C(L)\operatorname{Lip}(T)\le C(L).

References

Primary source

arXiv

Additional references

convex-analysisoptimal-transportprobability
Posted

Progress summary

Refreshed
Claimed solved

A September 2026 unrefereed preprint claims to settle the conjecture by extending dimension-free regularity estimates to arbitrary Lipschitz perturbations.

The conjecture asks whether the optimal transport map from a Gaussian measure to a log-Lipschitz perturbation remains globally Lipschitz with a dimension-independent bound. Fathi, Mikulincer, and Shenfeld stated it explicitly in work from 2023, published in 2024.

Known results

  • The conjecture is known in dimension one, where the heat-flow and Brenier maps coincide (Fathi, Mikulincer, Shenfeld, 2023).
  • The same work proves dimension-free bounds for a Langevin transport map, which generally differs from the Brenier map (Fathi, Mikulincer, Shenfeld, 2023).
  • Related Caffarelli estimates and interpolation results do not address this exact conjecture.

September 2026 claimed resolution

A September 2026 arXiv preprint, Caffarelli Estimates under Lipschitz Perturbations, claims to extend the differential estimates from pointwise Hessian comparisons to arbitrary globally Lipschitz perturbations and to obtain asymptotic, dimension-free constants. This is presented as resolving the conjecture, but the claim is unrefereed and unverified.

Current status (as of September 2026): The one-dimensional case and non-optimal transport estimates are established, while the broader Brenier-map conjecture is only claimed solved by an unverified preprint.

Sources

Solutions 0

No solutions have been posted yet.