Optimal Transport Wiki:Asymptotic equivalence of W 2 and H^-1

Asymptotic Equivalence of $W_{2}$ and $H^{-1}$.

Interest

When the quadratic Wasserstein and $H^{-1}$ distance become asymptotically equivalent when the involved densities are close to $\varrho =1$. This is particularly of interest in evolution problems with a constraint of $\varrho =1$ such as crowd motion.

Formalization

Lemma

Let $\mu ,\nu$ be absolutely continuous measures on a convex domain $\Omega$, with densities bounded by the same constant $C$. Then, for all function $\phi \in H^{1}(\Omega )$:

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

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

Definition of $H^{-1}$

The negative Sobolev norm $H^{-1}$ is defined:

$||\mu -\nu ||_{H^{-1}(\Omega )}={\text{sup}}{\Big \{}\int \phi d(\mu -\nu ):||\nabla \phi ||_{L^{2}}\leq 1{\Big \}}$

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

Proof of the Theorem uses the above Lemma and can be found Chapter 5, page 211 of .