  Benjamini, Haggstrom, Peres and Steif introduced the concept of a dynamical
random walk. This is a continuous family of random walks, {S_n(t)}. Benjamini
et. al. proved that if d=3 or d=4 then there is an exceptional set of t such
that {S_n(t)} returns to the origin infinitely often. In this paper we consider
a dynamical random walk on Z^2. We show that with probability one there exists
t such that {S_n(t)} never returns to the origin. This exceptional set of times
has dimension one. This proves a conjecture of Benjamini et. al.
