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

expository stuff to be filled in

Motivation

When the quadratic Wasserstein and ${\dot {H}}^{-1}$ 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 $C>0$. 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 .

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

The negative Sobolev norm ${\dot {H}}^{-1}$ is defined to be

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

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

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

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