arXiv 1401.70522014-01math.PRmath.ACmath.MG

度量测度空间半群及其无限可分概率测度

The semigroup of metric measure spaces and its infinitely divisible probability measures

Steven N. Evans, Ilya Molchanov

备注：: 48 pages, 0 figures. The previous version considered only compact metric measure spaces, but new arguments allow all the results of the previous version to be extended to the setting of complete separable metric measure spaces

摘要：

度量测度空间是一个完备的可分度量空间，它具有完备的概率测度。两个这样的空间是等价的，如果它们是等距的度量空间，通过等距映射第一个空间上的概率测度到第二个空间上的概率测度。等价类的集合可以用Greven，Pfaffelhuber和Winter的Gromov-Prohorov度量来度量。我们考虑自然二进制操作$\boxplus$在这个空间上，采取两个度量测度空间，并形成其笛卡尔积配备了两个度量的总和和两个概率测度的产品。我们表明，度量测度空间配备了这种操作形成一个可取消的，交换的，波兰半群的平移不变度量和每个元素有一个唯一的分解成素元素。我们调查的半群结构和自然行动的正实数在这个空间上，从缩放度量所产生的相互作用。例如，我们证明了对于任何给定的正实数$a,b,c$的平凡空间是唯一的空间$\mathcal{X}$满足$a \mathcal{X} \boxplus b \mathcal{X} = c \mathcal{X}$。我们确定不存在类似的大数定律：如果$\mathbf{X}_1, \mathbf{X}_2$.是随机空间的同分布独立序列，则没有$\frac{1}{n} \boxplus_{k=1}^n \mathbf{X}_k$的子序列按分布收敛，除非每个$\mathbf{X}_k$几乎必然等于平凡空间。刻画了该半群上的无穷可分概率测度和Lévy过程，刻画了稳定的概率测度，并建立了后者的LePage表示的对应.

英文摘要：

A metric measure space is a complete separable metric space equipped with probability measure that has full support. Two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure on the first space to the probability measure on the second. The resulting set of equivalence classes can be metrized with the Gromov-Prohorov metric of Greven, Pfaffelhuber and Winter. We consider the natural binary operation $\boxplus$ on this space that takes two metric measure spaces and forms their Cartesian product equipped with the sum of the two metrics and the product of the two probability measures. We show that the metric measure spaces equipped with this operation form a cancellative, commutative, Polish semigroup with a translation invariant metric and that each element has a unique factorization into prime elements. We investigate the interaction between the semigroup structure and the natural action of the positive real numbers on this space that arises from scaling the metric. For example, we show that for any given positive real numbers $a,b,c$ the trivial space is the only space $\mathcal{X}$ that satisfies $a \mathcal{X} \boxplus b \mathcal{X} = c \mathcal{X}$. We establish that there is no analogue of the law of large numbers: if $\mathbf{X}_1, \mathbf{X}_2$..., is an identically distributed independent sequence of random spaces, then no subsequence of $\frac{1}{n} \boxplus_{k=1}^n \mathbf{X}_k$ converges in distribution unless each $\mathbf{X}_k$ is almost surely equal to the trivial space. We characterize the infinitely divisible probability measures and the Lévy processes on this semigroup, characterize the stable probability measures and establish a counterpart of the LePage representation for the latter class.

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摘要：

英文摘要：

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