We introduce and study a class of random walks defined on the integer lattice -a discrete space and time counterpart of the symmetric α-stable process in . When 0 < α <2 any coordinate axis in , d ≥ 3, is a non-massive set whereas any cone is massive. We provide a necessary and sufficient condition for a thorn to be a massive set.
Let be a locally compact non-compact metric group. Assuming that is abelian we construct symmetric aperiodic random walks on with probabilities of return to any neighborhood V of the neutral element decaying at infinity almost as fast as the exponential function n ↦ exp(-n). We also show that for some discrete groups , the decay of the function can be made as slow as possible by choosing appropriate aperiodic random walks Sₙ on .
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