# Asymptotic equivalence of W 2 and H^-1

## Motivation

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

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

## Formalization

### Definition of ${\dot {H}}^{-1}$ The negative Sobolev norm $\|\cdot \|_{\dot {H}}^{-1}$ is defined   to be

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

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

$\int _{\Omega }\phi \,\mathrm {d} (\mu -\nu )\leq {\sqrt {C}}\|\nabla \phi \|_{L^{2}(\Omega )}W_{2}(\mu ,\nu )$ Proof of the lemma can be found Chapter 5, page 210 of .

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

This material is adapted from .

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

$\|\phi \|_{{\dot {H}}^{1}}:=\|\nabla \phi \|_{L^{2}(\Omega )}:=\left[\int _{\Omega }|\nabla \phi (x)|^{2}\,\mathrm {d} x\right]^{\frac {1}{2}}$ defines a semi-norm. Then for an absolutely continuous signed measure on $\Omega$ with zero total mass,

$\|\nu \|_{{\dot {H}}^{-1}}:=\sup \left\{|\langle \phi ,\nu \rangle |:\phi \in C^{1}(\Omega ),\,\|\phi \|_{{\dot {H}}^{1}}\leq 1\right\}=\sup \left\{\left|\int _{\Omega }\phi (x)\,\mathrm {d} \nu (x)\right|:\phi \in C^{1}(\Omega ),\,\|\phi \|_{{\dot {H}}^{1}}\leq 1\right\}.$ The space ${\dot {H}}^{-1}$ is the dual space of zero-mean $H^{1}(\Omega )$ functions endowed with the norm $L^{2}$ norm on the gradient.

### Theorem

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

$b^{-{\frac {1}{2}}}||\mu -\nu ||_{{\dot {H}}^{-1}(\Omega )}\leq W_{2}(\mu ,\nu )\leq a^{-{\frac {1}{2}}}||\mu -\nu ||_{{\dot {H}}^{-1}(\Omega )}$ The proof of the theorem uses the above lemma and can be found Chapter 5, page 211 of .

## Localization

The following material is adapted from .

This section deals with the problem of localization of the quadratic Wasserstein distance: if $\mu ,\nu$ are (signed) measures on $\mathbb {R} ^{d}$ that are close in the sense of $W_{2}$ , do they remain close to each other when restricted to subsets of $\mathbb {R} ^{d}$ ?

### Notation

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

• Recall that for a subset $A\subseteq \mathbb {R} ^{d}$ ,
$\mathrm {dist} (x,A):=\inf\{|x-y|:y\in A\}$ denotes the distance between a point $x$ and the subset $A$ .

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

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

### Theorem

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

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

$x\mapsto \mathrm {dist} (x,B^{c})^{2}$ on $B$ , that is, there exists constants $0 such that for all $x\in B$ ,

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

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

$W_{2}(a\varphi \cdot \mu ,\varphi \cdot \nu )\leq C(n)^{\frac {1}{2}}\left({\frac {c_{2}m_{2}}{c_{1}m_{1}}}\right)^{\frac {3}{2}}kc_{1}^{-{\frac {1}{2}}}W_{2}(\mu ,\nu ),$ for $C(n)<\infty$ some absolute constant depending only on $n$ . Moreover, taking $C(n):=2^{11}n$ fits. Furthermore, that $\varphi$ 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 .

## Connection with the Vlasov-Poisson Equation

Loeper  contributed an earlier result on a bound between $W_{2}$ and ${\dot {H}}^{-1}$ for bounded densities in studying the existence of solutions to the Vlasov-Poisson equation. Namely, Loeper proved that that if $\rho _{1},\rho _{2}$ be probability measures on $\mathbb {R} ^{d}$ with $L^{\infty }$ densities with respect to the Lebesgue measure. Let $\Psi _{i}$ , $i=1,2$ solve

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

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

$\|\nabla \Psi _{1}-\nabla \Psi _{2}\|_{L^{2}(\mathbb {R} ^{d})}\leq \left[\max \left\{\|\rho _{1}\|_{L^{\infty }},\|\rho _{2}\|_{L^{\infty }}\right\}\right]^{\frac {1}{2}}W_{2}(\rho _{1},\rho _{2}).$ Loeper also extended the result to finite measures with the same total mass.