U **petak, 13. rujna 2019. u 12:00 **u dvorani** B3-17 **PMF-a

**prof. Ilya Molchanov**

Institute of Mathematical Statistics and Actuarial Science, University of Bern

održat će predavanje

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

Predavanje je organizirano u sklopu redovitog kolokvija Znanstvenog razreda SMD-a.

Sažetak:

*The semigroup of metric measure spaces and its infinitely divisible probability measures
(joint work with Steve Evans, Berkeley)*

A metric measure space (also called Gromov triple) 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 with a translation invariant metric. There is an explicit family of continuous semicharacters that is extremely useful for establishing that there are no infinitely divisible elements 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. We establish that there is no analogue of the law of large numbers and 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.