Stochastic processes with independent increments on a groupare generalized to the concept of "white noise" on a Hopfalgebra or bialgebra. The main purpose of the book is thecharacterization of these processes as solutions of quantumstochastic differential equations in the sense of R.L.Hudsonand K.R. Parthasarathy. The notes are a contributionto quantum probability but they are also related toclassical probability, quantum groups, and operatoralgebras. The Az ma martingales appear as examples of whitenoise on a Hopf algebra which is a deformation of theHeisenberg group.The book will be of...