This is for continuous walk. As you say, in one dimension you will visit all the points. In two dimension the random walk is dense, so you pass arbitrarily cl

This is for continuous walk. As you say, in one dimension you will visit all the points. In two dimension the random walk is dense, so you pass arbitrarily close to every point in the plane. As the dimension increases beyond two, it gets less and less likely that you hit the target sphere. In some way, space gets bigger and bigger as the dimension increases. Suppose the d dimensional "target" sphere is very small and very close to the starting point of the path. Let some time pass, and suppose that the Brownian motion does not hit the sphere and has increased by 1 in every direction. Then the distance to the sphere is sqrt(d), which grows unbounded as d increases: if you don't hit the target immediately, on average you will be further away from it in higher dimension. So the probability of touching the target after that diminishes as the dimension increases. I don't have an intuitive explanation of why 3 is the critical dimension though.