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 $0<a<b<+\infty$. 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

$\exists 0<m_{1}\leq m_{2}<\infty \quad \forall x\in B\quad m_{1}\mathrm {d} \lambda (x)\leq \mathrm {d} \mu (x)\leq m_{2}\mathrm {d} \lambda (x).$

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 $0<c_{1}\leq c_{2}<\infty$ 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.