Save the Date: As part of the Mathematical Colloquium, a lecture by Ilya Molchanov (Bern) will take place on Friday, 28.06.2024 at 10:30 at the invitation of Prof. Dr. Evengy Spodarev.

Titel: Fundamental theorem of arithmetic for the semigroup of metric measure
spaces

Abstract: A metric measure space is a complete, separable metric space
equipped with a 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. We consider the natural binary operation 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, establishing that there are no
infinitely divisible elements and that each element has a unique
factorization
into prime elements. We establish that there is no analogue of the law of
large numbers for sequences of random metric measure spaces and characterize
the infinitely divisible probability measures and the Levy processes on this
semigroup, characterize the stable probability measures and establish a
counterpart of the LePage representation for the latter class.