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.