Under some mild condition, a random walk in the plane is recurrent. In
particular each trajectory is dense, and a natural question is how much time
one needs to approach a given small neighborhood of the origin. We address this
question in the case of some extended dynamical systems similar to planar
random walks, including \ZZ^2-extension of hyperbolic dynamics. We define a
pointwise recurrence rate and relate it to the dimension of the process, and
establish a convergence in distribution of the rescaled return times near the
origin