Suppose every point in the plane undergoes brownian motion for a time t. What is the probability n particles ended up at 0? For n finite, countable or uncountable?

What proportion of the plane does not have a particle on it after time t? Ie. pick n random points inside an open disc, as n approaches infinity, what fraction of those n points will have k particles on it?

Edit: Cut the plane into regions of equal area, let each region undergo brownian motion for t approaching infinity, what is the resulting distribution for the number of overlapping areas at origo?