Adam, I know that random walks on Z or Z^2 will, given enough steps, eventually hit any element with probability one, but not for Z^n, n>=3. Yet, I'm surprised to learn that 2D Brownian motion (say, on a disk, and which reflects when reaching the boundary) will cover a positive area with probability one. Can you point me to a reference, or did I misinterpret what you said? I would have guessed that with probability one, such a 2D Brownian motion will have 2D measure = 0, even after t = oo. --Dan On 2014-02-01, at 1:22 PM, Adam P. Goucher wrote:
Trace out a random walk (Brownian motion, but bounded within the confines of the galaxy), accelerating towards c so that the proper time elapsed is bounded. In particular, accelerate such that when the galaxy reaches an age of infinity, you have only aged by one hour.
Now, probabilistically you should collide infinitely often with absolutely everything, including your buddy, within one hour of proper time.
This also works in a two-dimensional unbounded universe, but not within a three-dimensional unbounded universe.