Transition probability estimates for long range random walks

Let (M,d, μ) be a uniformly discrete metric measure space satisfying space homogeneous volume doubling condition. We consider discrete time Markov chains on M symmetric with respect to μ and whose one-step transition density is comparable to (Vh(d(x, y))φ(d(x, y)) , where φ is a positive continuous regularly varying function with index β ∈ (0, 2) and Vh is the homogeneous volume growth function. Extending several existing work by other authors, we prove global upper and lower bounds for n-step transition probability density that are sharp up to constants.