Convergence of coined quantum walks in R d

Coined quantum walks may be interpreted as the motion in position space of a quantum particle with a spin degree of freedom; the dynamics are determined by iterating a unitary transformation which is the product of a spin transformation and a translation conditional on the spin state. Coined quantum walks on Z d can be treated as special cases of coined quantum walks on R d. We study quantum walks on R d and prove that the sequence of rescaled probability distributions in position space associated to the unitary evolution of the particle converges to a limit distribution.