Dilations of Markov Semigroups over Infinitely Divisible Convolution Semigroups

We report our results from Shalit and Skeide [SS20, Sections 14-16+Appendix E], about how to construct dilations of d–parameter Markov semigroups provided these strongly commute. In Bhat and Skeide [BS00] it became evident that an efficient way to con- struct dilations of one-parameter Markov (or more general CP-)semigroups, goes via product systems. The construction goes in two steps by, firstly, constructing for the Markov semigroup a product system (its GNS-product system) and, secondly, use this product system to construct a dilation. The two steps are very different in character; in particular, they depend on different properties that the semigroup R+ both possesses, but the semigroup Rd+ (d ≥ 2) does no longer possess both. The step that goes wrong is a general construction of d–parameter product system, while the construction of a dilation, once we have a product system, still works. So the main issue is to construct for d–parameter Markov semigroup a product system. However, Rd+ is the product of d semigroups R+, and for R+ we know how to construct a product system. The question is whether and how we can put together d one-parameter product systems to form a d–parameter product system. While for product systems of Hilbert spaces, for the one-parameter product systems of correspondences (as they occur from Markov semigroups), such a possibility does not exist. In [SS20], we give a complete answer (even for products of non-abelian semi- groups) for when (and how) we can form products of product systems. It turns out that in order to be able to apply this to the product systems of d commuting one- parameter semigroups, the latter have to commute strongly, a property introduced by Solel [Sol06]. We exploit this this to construct dilations of d–parameter Markov semigroups of classical L ́evy processes over Rn and of quantized d-parameter con- volution semigroups over Cn as introduced by Arveson [Arv02] and examined by Markiewicz [Mar03]. Due to some unexpected complications in the semigroup structure of the semi-group of infinitely divisible probability measures on Rn we were not yet able to dilation this whole semigroup