Recurrence of inhomogeneous random walks

Abstract

Inhomogeneous random walk means to study the time evolution of a sum of independent vector valued random variables (called “steps”), where we go beyond the traditional framework of Random Walks by dropping the usual condition of “identically distributed steps”. Our results show that genuinely 2-dimensional steps suffice—we do not need identically distributed steps to prove 2-dimensional Pólya type theorems. It turns out that we do not even need a whole 2-dimensional random walk: we can guarantee recurrence in a given sparse subsequence of the steps. An interesting example is the two-dimensional simple symmetric random walk on the lattice $\mathbb {Z}^2$, where we can prove recurrence for the sequence of primes; more precisely, for the sets $\{ p+1{:}\ p\ge 3\ \mathrm{prime}\}$, or $\{ p-1{:}\ p\ge 3\ \mathrm{prime}\}$, or $\{ 2p{:}\ p\ge 3\ \mathrm{prime}\}$, etc. (since for return we clearly need a set of even numbers). In this direction we can prove optimal results, describing the class of sparsest subsequences still exhibiting recurrence. In fact, we can prove these best possible results for the larger class of inhomogeneous random walks. We can also prove the one-dimensional analogs, which—not surprisingly—turn out to be easier.