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Long time behavior of random walks on abelian groups

Alexander Bendikov; Barbara Bobikau — 2010

Colloquium Mathematicae

Let be a locally compact non-compact metric group. Assuming that is abelian we construct symmetric aperiodic random walks on with probabilities $n \mapsto ℙ (S_{2 n} \in V)$ 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 $n \mapsto ℙ (S_{2 n} \in V)$ can be made as slow as possible by choosing appropriate aperiodic random walks Sₙ on .

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