Convergence theorems on multi-dimensional homogeneous quantum walks

We propose a general framework for quantum walks on d-dimensional spaces. We investigate asymptotic behavior of these walks. Among them, existence of limit distribution of homogeneous walks is proved. In this theorem, the support of the initial unit vector is not necessarily finite. We also pay attention on 1-cocycles. For homogeneous walks, convergence of averages of 1-cocycles associated to the position observable is also proved.