Categorial Independence and Lévy Processes

Symmetry Integrability and Geometry-Methods and Applications

article

Malte Gerhold¹  Stephanie Lachs¹  Michael Schürmann¹ 

[1] Institute of Mathematics and Computer Science, University of Greifswald;Department of Mathematical Sciences

关键词: general independence;    monoidal categories;    synthetic probability;    noncommutative probability;    quantum stochastic processes.;   

DOI  :  10.3842/SIGMA.2022.075

来源: National Academy of Science of Ukraine

【 摘 要 】

We generalize Franz' independence in tensor categories with inclusions from two morphisms (which represent generalized random variables) to arbitrary ordered families of morphisms. We will see that this only works consistently if the unit object is an initial object, in which case the inclusions can be defined starting from the tensor category alone. The obtained independence for morphisms is called categorial independence. We define categorial Lévy processes on every tensor category with initial unit object and present a construction generalizing the reconstruction of a Lévy process from its convolution semigroup via the Daniell-Kolmogorov theorem. Finally, we discuss examples showing that many known independences from algebra as well as from (noncommutative) probability are special cases of categorial independence.

【 授权许可】

Unknown