
This talk considers random objects that are infinitely
divisible in the sense of being representable as `sums' of other `independent'
random objects, where the terms in the decomposition are mutually similar
and the number of terms can be arbitrarily large. Based on concepts of
this type one may build a multitude of processes.The most well known such
concept is that of infinite divisibility in classical probability, with
the associated Levy processes. The study of this was founded by Kolmogorov,
Levy and Khintchine during the second quarter of the 20th Century and
has since grown into an enormously rich theory of great theoretical and
applied interest. The concept of infinitely divisible quantum instruments
is closely related to this. Recently, in the context of quantum stochastics,
several other types of infinite divisibility have arisen: "free,
monotone; antimonotone; boolean". Most developed at present is free
infinite divisibility which has its origin in the study of C* algebras
and the theory of large random matrices. The talk will survey the area
in not too technical terms.
