Brenier maps and convex potentials

Let μ and ν be probability measures on real d-space with finite second moments, and suppose μ is absolutely continuous with respect to d-dimensional Lebesgue measure. For cost c(x, y) = one-half times |x − y|², Brenier’s theorem gives a unique optimal coupling, of the form (id, T)#μ, with T the gradient of a convex potential ψ at μ-almost every source point. The target need not have a density; it may be atomic.

Absolute continuity of the source is a sufficient hypothesis, not a requirement imposed symmetrically on both marginals. Without a diffuse-source condition, splitting may be unavoidable, as a single source atom sent to two target atoms shows. The map is unique only up to μ-null sets. A potential can be altered by an additive constant, and its behavior away from the source’s relevant region need not be uniquely fixed. None of these qualifications weakens uniqueness of the actual optimal joint distribution.

At a smooth source point where the dual constraint is tight for a destination T(x), differentiating the contact relation in x yields gradient f(x) = x − T(x). Thus T is the gradient of |x|²/2 − f(x). The c-concavity structure of an optimal quadratic potential makes this new function convex. This differentiable calculation is motivation; the full theorem uses almost-everywhere differentiability and convex analysis rather than assuming all data are smooth.

Convex gradients are monotone: their differences have nonnegative inner product with differences of the inputs. In one dimension this is the familiar nondecreasing rearrangement. In higher dimensions, ordinary two-point monotonicity is not sufficient to characterize all convex gradients; cyclic monotonicity supplies the stronger structure. A nonsymmetric linear shear is not the gradient of a scalar convex quadratic, even though it can be invertible and can push one smooth distribution to another.

If source and target have smooth positive densities and the optimal gradient map is sufficiently differentiable and locally invertible, change of variables gives target density evaluated at ∇ψ times det(D²ψ) equal to the source density. This is the Monge–Ampère equation. Those regularity and invertibility assumptions are additional: Brenier’s basic conclusion alone does not guarantee a classical Hessian or a smooth solution of that equation.

For practical calculations, first prove that a proposed convex-gradient map has the correct pushforward and finite cost. These facts certify optimality under the theorem’s setting without solving a nonlinear PDE numerically. Scaling a one-dimensional uniform distribution is a transparent example. Gaussian covariance examples similarly reduce to linear algebra when the relevant map is a symmetric positive matrix. Keep the one-half convention visible: the optimal coupling is unchanged if the cost is doubled, but the reported minimum value doubles.