Re: [math-fun] physics thought experiment
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.
At t = infinity, the path forms a dense subset of R^2 (i.e. impregnates every open ball). This is not the same as being a space-filling curve (it could have measure 0, for instance), but does pass within every epsilon > 0 of every point. This is equivalent to a two-dimensional random walk on Z^2 almost surely reaching every point, which is in turn equivalent to a two-dimensional random walk on Z^2 being recurrent (almost surely returning to the origin infinitely often). Sincerely, Adam P. Goucher
--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.
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Aha -- that makes sense, thank you. (In fact a Brownian path in R^2 will with probability one have zero Lebesgue 2D measure.) --Dan On 2014-02-02, at 6:32 AM, Adam P. Goucher wrote:
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.
At t = infinity, the path forms a dense subset of R^2 (i.e. impregnates every open ball). This is not the same as being a space-filling curve (it could have measure 0, for instance), but does pass within every epsilon > 0 of every point.
This is equivalent to a two-dimensional random walk on Z^2 almost surely reaching every point, which is in turn equivalent to a two-dimensional random walk on Z^2 being recurrent (almost surely returning to the origin infinitely often).
participants (2)
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Adam P. Goucher -
Dan Asimov