Evolution equations and Lévy processes on quantum groups

Abstract

In this chapter we consider stochastic processes on quantum groups that are related to evolution equations of the form $$ {\partial _t}u = Lu$$ with some difference-differential operator L. For the equations considered in Section 7.1, u is an element of a quantum or braided group A. We recall that solutions of these equations can be given as Appell systems or shifted moments of the associated process, and show how these can be calculated explicitly on the q-affine group and on a braided analogue of the Heisenberg-Weyl group.