Restoring Wasserstein Rigidity with a single point
We consider isometrically flexible Wasserstein spaces and demonstrate that adding a single point to the underlying metric space makes these Wasserstein spaces rigid.
math.MG↗
arXiv subjects
Eric Ströher
Publications and source records attributed to Eric Ströher.
Wasserstein Rigidity over $\mathbb{R}^n$ with smooth norms
We study $p-$Wasserstein spaces $ \mathcal{W}_p(\mathbb{R}^n, d_N)$ over $\mathbb{R}^n$ equipped with a norm metric $d_N$. We show that, if the norm is smooth enough, then the Wasserstein space is isometrically rigid whenever $p \neq 2$. We also show that, even when $p=2$, we can recover the isometric rigidity of the Wasserstein space $\mathcal{W}_2(\mathbb{R}^n, d_N)$ when $N$ is an $l_q-$norm and $q>2$.
math.MG↗
Rigid and flexible Wasserstein spaces
In this paper, we study isometries of $p$-Wasserstein spaces. In our first result, for every complete and separable metric space $X$ and for every $p\geq1$, we construct a metric space $Y$ such that $X$ embeds isometrically into $Y$, and the $p$-Wasserstein space over $Y$ admits mass-splitting isometries. Our second result is about embeddings into rigid constructions. We show that any complete and separable metric space $X$ can be embedded isometrically into a metric space $Y$ such that the $1$-Wasserstein space is isometrically rigid.
math.MG↗
Sommerfeld Radiation Condition for Helmholtz Equations with long-range Potentials
We study the electric Helmholtz equation $Δu + Vu + λu =f$ and show that, for certain potentials, the solution $u$ given by the limited absorption principle obeys a Sommerfeld radiation condition. We use a non-spherical approach based on the solution $K$ of the eikonal equation $|\nabla K|^2=1 + \frac{p}λ$ to improve previous results in that area and extend them to long-range potentials which decay like $|x|^{-2-α}$ at infinity, with $α> 0$.
math.AP↗