Transition probability estimates for long range random walks

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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))⁻¹, 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.