Recently Kloeckner described the structure of the isometry group of the quadratic Wasserstein space $\mathcal{W}_2(\R^n)$. 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 $\isom\ler{\Wp(\R)}$, the isometry group of the Wasserstein space $\Wp(\R)$ for all $p \in [1, \infty)\setminus\{2\}$. We show that $\mathcal{W}_2(\R)$ is also exceptional regarding the parameter $p$: $\Wp(\R)$ 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 $\R$ by the compact interval $[0,1]$. Surprisingly, in that case, $\Wp\zo$ is isometrically rigid if and only if $p\neq1$. Moreover, $\Wo\zo$ admits isometries that split mass, and $\isom\ler{\Wo\zo}$ cannot be embedded into $\isom\ler{\wor}$.

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Wasserstein 空间的等距研究——实线

最近Kloeckner描述了二次Wasserstein空间$\mathcal{W}_2(\R^n)$的等距群的结构。事实证明，在存在奇异等距流的意义上，实线的情况是例外的。沿着这条调查线，我们计算 $\isom\ler{\Wp(\R)}$，Wasserstein 空间 $\Wp(\R)$ 对于所有 $p \in [1, \infty) 的等距群\setminus\{2\}$. 我们证明 $\mathcal{W}_2(\R)$ 对于参数 $p$ 也是例外的：当且仅当 $p\neq 2$ 时，$\Wp(\R)$ 是等距刚性的。关于底层空间，我们证明如果我们用紧致区间 $[0,1]$ 替换 $\R$，$p=2$ 的例外性消失了。令人惊讶的是，在这种情况下，$\Wp\zo$ 是等距刚性的当且仅当 $p\neq1$。此外，$\Wo\zo$ 承认分裂质量的等距，