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

Abstract

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 L\'evy processes on this semigroup, characterize the stable probability measures and establish a counterpart of the LePage representation for the latter class.