Quantum diffusions and Appell systems

Within the algebraic framework of Hopf algebras, random walks and associated diffusion equations (master equations) are constructed and studied for two basic operator algebras of quantum mechanics i.e. the Heisenberg–Weyl algebra (hw) and its q-deformed version hwq. This is done by means of functionals determined by the associated coherent state density operators. The ensuing master equations admit solutions given by hw and hwq-valued Appell systems.