# Asymptotic equivalence of W 2 and H^-1

## Motivation

The quadratic Wasserstein distance and ${\displaystyle {\dot {H}}^{-1}}$ distance become asymptotically equivalent when the when the measures are absolutely continuous with respect to Lebesgue measure with density close to the value ${\displaystyle \varrho =1}$. This is particularly of interest since the space ${\displaystyle H^{-1}}$ is a Hilbert space as opposed to ${\displaystyle W_{2}}$ being only a metric space. This allows one to extend several well-known results about continuity of various operators in ${\displaystyle H^{-1}}$ to ${\displaystyle W_{2}}$ by asymptotic equivalence. This equivalence is also important numerically, where computing ${\displaystyle H^{-1}}$ is much easier than computing ${\displaystyle W_{2}}$.

Furthermore, this asymptotic equivalence is relevant for evolution problems with the constraint ${\displaystyle \varrho \leq 1}$, such as crowd motion. [1]

## Formalization

### Definition of ${\displaystyle {\dot {H}}^{-1}}$

The negative Sobolev norm ${\displaystyle \|\cdot \|_{\dot {H}}^{-1}}$ is defined [1] [2] to be

${\displaystyle \|\mu -\nu \|_{{\dot {H}}^{-1}(\Omega )}:=\sup \left\{\int _{\Omega }\phi \,\mathrm {d} (\mu -\nu ):\phi \in C_{c}^{\infty }(\Omega ),\,\|\nabla \phi \|_{L^{2}(\Omega )}\leq 1\right\}.}$


### Lemma

Let ${\displaystyle \mu ,\nu }$ be measures that are absolutely continuous with respect to Lebesgue measure on a convex domain ${\displaystyle \Omega }$, with densities bounded above by the same constant ${\displaystyle C>0}$. Then, for all functions ${\displaystyle \phi \in H^{1}(\Omega )}$:

${\displaystyle \int _{\Omega }\phi \,\mathrm {d} (\mu -\nu )\leq {\sqrt {C}}\|\nabla \phi \|_{L^{2}(\Omega )}W_{2}(\mu ,\nu )}$


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

### ${\displaystyle {\dot {H}}^{-1}}$ as a Dual

This material is adapted from [3].

An important property of ${\displaystyle {\dot {H}}^{-1}}$ is its characterization as a dual, which justifies the notation. Let ${\displaystyle \Omega \subseteq \mathbb {R} ^{d}}$ be an open and connected subset. For ${\displaystyle \phi \in C^{1}(\Omega )}$,

${\displaystyle \|\phi \|_{{\dot {H}}^{1}}:=\|\nabla \phi \|_{L^{2}(\Omega )}:=\left[\int _{\Omega }|\nabla \phi (x)|^{2}\,\mathrm {d} x\right]^{\frac {1}{2}}}$


defines a semi-norm. Then for an absolutely continuous signed measure on ${\displaystyle \Omega }$ with zero total mass,

${\displaystyle \|\nu \|_{{\dot {H}}^{-1}}:=\sup \left\{|\langle \phi ,\nu \rangle |:\phi \in C^{1}(\Omega ),\,\|\phi \|_{{\dot {H}}^{1}}\leq 1\right\}=\sup \left\{\left|\int _{\Omega }\phi (x)\,\mathrm {d} \nu (x)\right|:\phi \in C^{1}(\Omega ),\,\|\phi \|_{{\dot {H}}^{1}}\leq 1\right\}.}$


The space ${\displaystyle {\dot {H}}^{-1}}$ is the dual space of zero-mean ${\displaystyle H^{1}(\Omega )}$ functions endowed with the norm ${\displaystyle L^{2}}$ norm on the gradient.

### Theorem

Let ${\displaystyle \mu ,\nu }$ be absolutely continuous measures on a convex domain ${\displaystyle \Omega }$, with densities bounded from below and from above by the same constants ${\displaystyle a,b}$ with ${\displaystyle 0. Then

${\displaystyle b^{-{\frac {1}{2}}}||\mu -\nu ||_{{\dot {H}}^{-1}(\Omega )}\leq W_{2}(\mu ,\nu )\leq a^{-{\frac {1}{2}}}||\mu -\nu ||_{{\dot {H}}^{-1}(\Omega )}}$


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 ${\displaystyle \mu ,\nu }$ are (signed) measures on ${\displaystyle \mathbb {R} ^{d}}$ that are close in the sense of ${\displaystyle W_{2}}$, do they remain close to each other when restricted to subsets of ${\displaystyle \mathbb {R} ^{d}}$?

### Notation

Here we are working in Euclidean space ${\displaystyle \mathbb {R} ^{d}}$ with the Lebesgue measure ${\displaystyle \lambda }$.

• Recall that for a subset ${\displaystyle A\subseteq \mathbb {R} ^{d}}$,
${\displaystyle \mathrm {dist} (x,A):=\inf\{|x-y|:y\in A\}}$

denotes the distance between a point ${\displaystyle x}$ and the subset ${\displaystyle A}$.

• For a (signed) measure ${\displaystyle \mu }$ on ${\displaystyle \mathbb {R} ^{d}}$ and ${\displaystyle \varphi :\mathbb {R} ^{d}\to \mathbb {R} }$ a nonnegative and measurable function, ${\displaystyle \varphi \cdot \mu }$ denotes the measure such that ${\displaystyle \mathrm {d} (\varphi \cdot \mu )=\varphi (x)\,\mathrm {d} \mu (x)}$.
• The norm
${\displaystyle \|\mu \|_{1}:=\int _{\mathbb {R} ^{d}}\,|\mathrm {d} \mu (x)|}$

denotes the total variation norm of the signed measure ${\displaystyle \mu }$. If ${\displaystyle \mu }$ is in fact a measure, then ${\displaystyle \|\mu \|_{1}=\mu (\mathbb {R} ^{d})}$.

Now we can ask the original question more precisely. If ${\displaystyle \varphi :\mathbb {R} ^{d}\to \mathbb {R} }$ is non-negative and compactly supported satisfying further technical assumptions to be specified later, we wish to bound ${\displaystyle W_{2}(a\varphi \cdot \mu ,\varphi \cdot \nu )}$ by ${\displaystyle W_{2}(\mu ,\nu )}$, where ${\displaystyle a}$ is a constant factor ensuring that ${\displaystyle a\varphi \cdot \mu }$ and ${\displaystyle \varphi \cdot \nu }$ have the same mass. The factor of ${\displaystyle a}$ is necessary, otherwise the ${\displaystyle W_{2}}$ distance between ${\displaystyle \varphi \cdot \mu }$ and ${\displaystyle \varphi \cdot \nu }$ is in general not well-defined.

### Theorem

Let ${\displaystyle \mu ,\nu }$ be measures on ${\displaystyle \mathbb {R} ^{d}}$ having the same total mass, and let ${\displaystyle B}$ be a ball in ${\displaystyle \mathbb {R} ^{d}}$. Assume that on ${\displaystyle B}$, the density of ${\displaystyle \mu }$ with respect to the Lebesgue measure is bounded above and below, that is

${\displaystyle \exists 0

Let ${\displaystyle \varphi :\mathbb {R} ^{d}\to (0,+\infty )}$ be a ${\displaystyle k}$-Lipschitz function for some ${\displaystyle 0\leq k<\infty }$ supported in ${\displaystyle B}$, and suppose that ${\displaystyle \varphi }$ is bounded above and below by the map

${\displaystyle x\mapsto \mathrm {dist} (x,B^{c})^{2}}$

on ${\displaystyle B}$, that is, there exists constants ${\displaystyle 0 such that for all ${\displaystyle x\in B}$,

${\displaystyle c_{1}\mathrm {dist} (x,B^{c})^{2}\leq \varphi (x)\leq c_{2}\mathrm {dist} (x,B^{c})^{2}.}$

Then, denoting

${\displaystyle a:=\|\varphi \cdot \nu \|_{1}/\|\varphi \cdot \mu \|_{1}={\frac {\int _{\mathbb {R} ^{d}}|\varphi (x)\,\mathrm {d} \mu (x)|}{\int _{\mathbb {R} ^{d}}|\varphi (x)\,\mathrm {d} \nu (x)|}},}$

we have

${\displaystyle W_{2}(a\varphi \cdot \mu ,\varphi \cdot \nu )\leq C(n)^{\frac {1}{2}}\left({\frac {c_{2}m_{2}}{c_{1}m_{1}}}\right)^{\frac {3}{2}}kc_{1}^{-{\frac {1}{2}}}W_{2}(\mu ,\nu ),}$


for ${\displaystyle C(n)<\infty }$ some absolute constant depending only on ${\displaystyle n}$. Moreover, taking ${\displaystyle C(n):=2^{11}n}$ fits. Furthermore, that ${\displaystyle \varphi }$ 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 ${\displaystyle W_{2}}$ and ${\displaystyle {\dot {H}}^{-1}}$ for bounded densities in studying the existence of solutions to the Vlasov-Poisson equation. Namely, Loeper proved that that if ${\displaystyle \rho _{1},\rho _{2}}$ be probability measures on ${\displaystyle \mathbb {R} ^{d}}$ with ${\displaystyle L^{\infty }}$ densities with respect to the Lebesgue measure. Let ${\displaystyle \Psi _{i}}$, ${\displaystyle i=1,2}$ solve

${\displaystyle -\Delta \Psi _{i}=\rho _{i}\qquad {\text{in }}\mathbb {R} ^{d},}$
${\displaystyle \Psi _{i}(x)\to 0\qquad {\text{as }}|x|\to \infty ,}$

in the integral sense, that is,

${\displaystyle \Psi _{i}(x)={\frac {1}{4\pi }}\int _{\mathbb {R} ^{d}}{\frac {\rho _{i}(y)}{|x-y|}}\,\mathrm {d} y.}$

Then

${\displaystyle \|\nabla \Psi _{1}-\nabla \Psi _{2}\|_{L^{2}(\mathbb {R} ^{d})}\leq \left[\max \left\{\|\rho _{1}\|_{L^{\infty }},\|\rho _{2}\|_{L^{\infty }}\right\}\right]^{\frac {1}{2}}W_{2}(\rho _{1},\rho _{2}).}$


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

## References

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.
2. [2] Peyre, Rémi. Comparison between ${\displaystyle W_{2}}$ distance and ${\displaystyle {\dot {H}}^{-1}}$ norm, and localisation of Wasserstein distance.