In  Benjamini, Haggstrom, Peres and Steif introduced the model of dynamical random walk on the $d$-dimensional lattice $Z^d$. This is a continuum of random walks indexed by a time parameter $t$. They proved that for dimensions $d=3,4$ there almost surely exist times $t$ such that the random walk at time $t$ visits the origin infinitely often, but for dimension 5 and up there almost surely do not exist such $t$. Hoffman showed that for dimension 2 there almost surely exists $t$ such that the random walk at time $t$ visits the origin only finitely many times . We refine the results of  for dynamical random walk on $Z^2$, showing that with probability one the are times when the origin is visited only a finite number of times while other points are visited infinitely often.
"A special set of exceptional times for dynamical random walk on $Z^2$." Electron. J. Probab. 13 1927 - 1951, 2008. https://doi.org/10.1214/EJP.v13-571