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?



Brownian motion of every point in the plane
Presumably you mean you have continuum-many independent Brownian motions, one (call it $W\_p(t)$) with $W\_p(0) = p$ for each $p$. Unfortunately I'm pretty sure the number of $p$ for which $W\_p(t) = 0$ is not a measurable function, so your 
question does not have an answer. 


