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If I start a random walk in an $n$-dimensional box , say $[0,1]^n$, with reflective boundaries (i.e. the random walk is never permitted to leave the box), will its orbit eventually be dense in the box? Is there a theorem that says this is true/false? I feel like this is true for $n=1$ or $n=2$, but what about in higher dimensions, say, $n=4$ or $n=5$? Any references would be appreciated.

Edit: I was sloppy; my apologies. I said random walk because I'm implementing this in code and necessarily the increments are not continuous. I would be happy with an answer related to Brownian motion. This question has empirical research relevance for me: I'm exploring a set whose shape I don't know by letting a random "particle" walk around inside it. I want to know if it will trace out the entire set if in theory I let the program run long enough. (Thanks for the comments.)

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    $\begingroup$ This question is probably better suited for math.stackexchange.com as it is not really research level. Also, "random walk" usually refers to a process moving in discrete space; perhaps you are thinking of Brownian motion? $\endgroup$
    – Nate Eldredge
    Commented Feb 10, 2012 at 0:58
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    $\begingroup$ If the question is asked on MSE then it should give more details about the tacit assumptions, as per Nate Eldredge's comments. Without specifying the law of your stochastic process the question is ill-posed $\endgroup$
    – Yemon Choi
    Commented Feb 10, 2012 at 1:56
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    $\begingroup$ @covstat, are you sure you want a random walk and not a billiard path? $\endgroup$
    – JRN
    Commented Feb 10, 2012 at 2:49
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    $\begingroup$ I edited the question. $\endgroup$
    – covstat
    Commented Feb 10, 2012 at 15:53
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    $\begingroup$ The random path up to a given finite time is not dense in the cube. The full random path is (almost surely) dense in the cube, in every dimension. This is basically a consequence of Kolmogorov zero-one law: for every tiny part $S$ of the box, the path of time-length $1$ visits $S$ with probability at least $u>0$, uniformly over its starting point. Iterating this, one sees that $S$ is never visited before time $n$ with probability at most $(1-u)^n\to0$. $\endgroup$
    – Did
    Commented Feb 10, 2012 at 16:06

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