Asymptotic equivalence of W 2 and H^-1

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The quadratic Wasserstein distance and distance become asymptotically equivalent when the when the measures are absolutely continuous with respect to Lebesgue measure with density close to the value . This is particularly of interest since the space is a Hilbert space as opposed to being only a metric space. This allows one to extend several well-known results about continuity of various operators in to by asymptotic equivalence. This equivalence is also important numerically, where computing is much easier than computing .

Furthermore, this asymptotic equivalence is relevant for evolution problems with the constraint , such as crowd motion. [1]


Definition of

The negative Sobolev norm is defined [1] [2] to be


Let be measures that are absolutely continuous with respect to Lebesgue measure on a convex domain , with densities bounded above by the same constant . Then, for all functions :

Proof of the lemma can be found Chapter 5, page 210 of [1].

as a Dual

This material is adapted from [3].

An important property of is its characterization as a dual, which justifies the notation. Let be an open and connected subset. For ,

defines a semi-norm. Then for an absolutely continuous signed measure on with zero total mass,

The space is the dual space of zero-mean functions endowed with the norm norm on the gradient.


Let be absolutely continuous measures on a convex domain , with densities bounded from below and from above by the same constants with . Then

The proof of the theorem uses the above lemma and can be found Chapter 5, page 211 of [1].


The following material is adapted from [3].

This section deals with the problem of localization of the quadratic Wasserstein distance: if are (signed) measures on that are close in the sense of , do they remain close to each other when restricted to subsets of ?


Here we are working in Euclidean space with the Lebesgue measure .

  • Recall that for a subset ,

denotes the distance between a point and the subset .

  • For a (signed) measure on and a nonnegative and measurable function, denotes the measure such that .
  • The norm

denotes the total variation norm of the signed measure . If is in fact a measure, then .

Now we can ask the original question more precisely. If is non-negative and compactly supported satisfying further technical assumptions to be specified later, we wish to bound by , where is a constant factor ensuring that and have the same mass. The factor of is necessary, otherwise the distance between and is in general not well-defined.


Let be measures on having the same total mass, and let be a ball in . Assume that on , the density of with respect to the Lebesgue measure is bounded above and below, that is

Let be a -Lipschitz function for some supported in , and suppose that is bounded above and below by the map

on , that is, there exists constants such that for all ,

Then, denoting

we have

for some absolute constant depending only on . Moreover, taking fits. Furthermore, that is supported in a ball is not necessary, as it can be supported in a cube or a simplex.

The proof can be found in [3].

Connection with the Vlasov-Poisson Equation

Loeper [2] contributed an earlier result on a bound between and for bounded densities in studying the existence of solutions to the Vlasov-Poisson equation. Namely, Loeper proved that that if be probability measures on with densities with respect to the Lebesgue measure. Let , solve

in the integral sense, that is,


Loeper also extended the result to finite measures with the same total mass.


  1. 1.0 1.1 1.2 1.3 F. Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 5, pages 209-211
  2. 2.0 2.1 [1] Loeper, Grégoire. Uniqueness of the solution to the Vlasov–Poisson system with bounded density. Journal de Mathématiques Pures et Appliquées, Volume 86, Issue 1, 2006, Pages 68-79, ISSN 0021-7824.
  3. 3.0 3.1 3.2 [2] Peyre, Rémi. Comparison between distance and norm, and localisation of Wasserstein distance.