Long range random walks and associated geometries on groups of polynomial growth

In the context of countable groups polynomial volume growth, we consider a large class random walks that are allowed to take long jumps along multiple subgroups according power law distributions. For such walk, study time behavior its probability return at n in terms key parameters describing driving measure and structure underlying group. We obtain assorted estimates including near-diagonal two-sided Hölder continuity solutions associated discrete parabolic difference equation. each case, these involve construction geometry adapted walk.