Consider a simple random walk on $\mathbb{Z}^d$, $d \geq 3$, which starts from the origin. As $d \geq 3$, there is a positive probability that the random walk never visits the origin again.
Now, let a new simple random start from any of the vertices which have been visited by the first random walk. Is the probability that no of the new random walks visits the origin positive as well?
Perhaps the answer could be positive for large dimensions, but I am not sure.