Transition Probabilities for Symmetric Jump Processes

Abstract

We consider symmetric Markov chains on the integer lattice in d d dimensions, where α ∈ ( 0 , 2 ) \alpha \in (0,2) and the conductance between x x and y y is comparable to | x − y | − ( d + α ) |x-y|^{-(d+\alpha )} . We establish upper and lower bounds for the transition probabilities that are sharp up to constants.