On massive sets for subordinated random walks

Abstract

We study massive (reccurent) sets with respect to a certain random walk $S_{\alpha }$ defined on the integer lattice ${\mathbb{Z}}^d$, $d=1,2$. Our random walk $S_{\alpha }$ is obtained from the simple random walk $S$ on ${\mathbb{Z}}^d$ by the procedure of discrete subordination. $S_{\alpha }$ can be regarded as a discrete space and time counterpart of the symmetric $\alpha$-stable Lévy process in ${\mathbb{R}}^d$. In the case $d=1$ we show that some remarkable proper subsets of $\mathbb{Z}$ , e.g. the set $\mathcal{P}$ of primes, are massive whereas some proper subsets of $\mathcal{P}$ such as Leitmann primes ${\mathcal{P}}_h$ are massive/non-massive depending on the function $h$. Our results can be regarded as an extension of the results of McKean (1961) about massiveness of the set of primes for the simple random walk in ${\mathbb{Z}}^3$. In the case $d=2$ we study massiveness of thorns and their proper subsets.

16 pages, 1 figure

References in corpus (1)

• Nash type inequalities for fractional powers of non-negative self-adjoint operators

Cited by in corpus (1)

• Stationary entrance chains and applications to random walks