论文标题
Wasserstein空间的等距研究---真实线
Isometric study of Wasserstein spaces --- the real line
论文作者
论文摘要
最近，克洛克纳（Kloeckner）描述了二次Wasserstein Space $ \ Mathcal {w} _2 \ left（\ Mathbb {r}^n \ right）$的结构。事实证明，实际线的情况是有异国情调的等距流的意义上的特殊情况。在此调查之后，我们计算$ \ mathrm {isom} \ left（\ Mathcal {w} _p（\ Mathbb {r}）\ right）$，wasserstein space $ \ nathcal $ \ Mathcal {w} w} _p（w} \ infty）\ setMinus \ {2 \} $。我们表明，$ \ MATHCAL {W} _2（\ Mathbb {r}）$在参数$ p $：$ \ mathcal {w} _p（\ Mathbb {r}）$上是异常的，如果$ P \ neq 2 $。关于基础空间，我们证明，如果我们用紧凑型间隔$ [0,1] $替换$ \ mathbb {r} $，$ p = 2 $的异常性消失。令人惊讶的是，在这种情况下，$ \ MATHCAL {W} _p \ left（[0,1] \右）$在且仅当$ p \ neq1 $时是等值固定的。此外，$ \ MATHCAL {W} _1 \ left（[0,1] \ right）$允许分裂质量的异构体，$ \ mathrm {isom} \ left（\ Mathcal {W} _1 _1 _1 _1 \ left（[0,1] \ right） $ \ mathrm {isom} \ left（\ Mathcal {w} _1（\ Mathbb {r}）\ right）。$
Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space $\mathcal{W}_2\left(\mathbb{R}^n\right)$. It turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow. Following this line of investigation, we compute $\mathrm{Isom}\left(\mathcal{W}_p(\mathbb{R})\right)$, the isometry group of the Wasserstein space $\mathcal{W}_p(\mathbb{R})$ for all $p \in [1, \infty)\setminus\{2\}$. We show that $\mathcal{W}_2(\mathbb{R})$ is also exceptional regarding the parameter $p$: $\mathcal{W}_p(\mathbb{R})$ is isometrically rigid if and only if $p\neq 2$. Regarding the underlying space, we prove that the exceptionality of $p=2$ disappears if we replace $\mathbb{R}$ by the compact interval $[0,1]$. Surprisingly, in that case, $\mathcal{W}_p\left([0,1]\right)$ is isometrically rigid if and only if $p\neq1$. Moreover, $\mathcal{W}_1\left([0,1]\right)$ admits isometries that split mass, and $\mathrm{Isom}\left(\mathcal{W}_1\left([0,1]\right)\right)$ cannot be embedded into $\mathrm{Isom}\left(\mathcal{W}_1(\mathbb{R})\right).$
