Asymptotic equivalence of W2 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
The negative Sobolev norm \(\| \cdot \|_\dot{H}^{-1}\) is defined to be
\(\| \mu - \nu \|_{\dot{H}^{-1} (\Omega)} := \sup \left \{ \int_{\Omega} \phi \, \mathrm{d}( \mu - \nu ) : \phi \in C^{\infty}_c(\Omega), \, \| \nabla \phi \|_{L^2(\Omega)} \leq 1 \right \} .\)
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 .
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
\(W_2 (a\varphi \cdot \mu , \varphi \cdot \nu) \leq C(n)^{\frac{1}{2}} \left( \frac{ c_2 m_2 }{ c_1 m_1 } \right)^{\frac{3}{2}} k c_1^{-\frac{1}{2}} W_2(\mu,\nu) ,\)
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.
References
F. Santambrogio, Optimal Transport for Applied Mathematicians, Chapter 5, pages 209-211