  Benjamini,Haggstrom, Peres and Steif introduced the model of dynamical random
walk on Z^d. This is a continuum of random walks indexed by a parameter t. They
proved that for d=3,4 there almost surely exist t such that the random walk at
time t visits the origin infinitely often, but for d > 4 there almost surely do
not exist such t. Hoffman showed that for d=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 Hoffman for dynamical random walk on Z^2, showing that
with probability one there are times when the origin is visited only a finite
number of times while other points are visited infinitely often.
