  Working withing the framework of Hopf algebras, a random walk and the
associated diffusion equation are constructed on a space that is algebraically
described as the merging of the real line algebra with the anyonic line
algebra. Technically this merged structure is a smash algebra, namely an
algebra resulting by a braided tensoring of real with anyonic line algebras.
  The motivation of introducing the smashing results from the necessity of
having non commuting increments in the random walk. Based on the
observable-state duality provided by the underlying Hopf structure, the
construction is cast into two dual forms: one using functionals determined by
density probability functions and the other using the associated Markov
transition operator. The ensuing diffusion equation is shown to possess
triangular matrix realization.
  The study is completed by the incorporation of Hamiltonian dynamics in the
above random walk model, and by the construction of the dynamical equation
obeyed by statistical moments of the problem for generic entangled density
functions.
