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.

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.)