Motivation
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]
Formalization
Definition of 
The negative Sobolev norm
is defined [1] [2] to be
Lemma
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.
Theorem
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].
Localization
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
?
Notation
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.
Theorem
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,

Then
Loeper also extended the result to finite measures with the same total mass.
References
- ↑ 1.0 1.1 1.2 1.3 F. Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 5, pages 209-211
- ↑ 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.
- ↑ 3.0 3.1 3.2 [2] Peyre, Rémi. Comparison between
distance and
norm, and localisation of Wasserstein distance.