We describe basic motivations behind quantum or noncommutative probability, introduce quantum Lévy processes on compact quantum groups, and discuss several aspects of the study of the latter in the example of quantum permutation groups. The first half of this paper is a survey on quantum probability, compact quantum groups, and Lévy processes on compact quantum groups. In the second half the theory is applied to quantum permutations groups. Explicit examples are constructed and certain classes of such Lévy processes are classified.
Franz, U., Kula, A., & Skalski, A. (2016). Lévy processes on quantum permutation groups. In Operator Theory: Advances and Applications (Vol. 252, pp. 193–259). Springer International Publishing. https://doi.org/10.1007/978-3-319-29116-1_11