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α-stable random walk has massive thorns

Alexander Bendikov; Wojciech Cygan — 2015

Colloquium Mathematicae

We introduce and study a class of random walks defined on the integer lattice $ℤ^{d}$ -a discrete space and time counterpart of the symmetric α-stable process in $ℝ^{d}$ . When 0 < α <2 any coordinate axis in $ℤ^{d}$ , 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.

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 .

Invariant local Dirichlet forms on locally compact groups

Alexander Bendikov; Laurent Saloff-Coste — 2002

Annales de la Faculté des sciences de Toulouse : Mathématiques

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Currently displaying 1 – 3 of 3

α-stable random walk has massive thorns

Long time behavior of random walks on abelian groups

Invariant local Dirichlet forms on locally compact groups