So one of my favorite mathematical quotes is one that Rick Durrett (in his text Probability: Theory and Examples) credits to Shizuo Kakutani: "A drunk man will eventually find his way home but a drunk bird may get lost forever." More formally, random walks on the square lattice in two dimensions return to the origin infinitely often; random walks in two dimensions with more general steps allowed, but in which the expected position at any time is still zero, return arbitrarily close to the origin infinitely often. In three dimensions this is not true. (Random walks that return arbitrarily close to the origin infinitely often are called "recurrent", the others "transient".)
I just came across this paper from MIT's Undergraduate Seminar in Discrete Mathematics (18.304, Spring 2006), for which an anonymous student wrote some notes on simple random walks. Here we learn that a drunk man will eventually get home, but a drunk man who has had drinks containing Red Bull will not; as you know from the commercials, "Red Bull gives you wings!"
Another random walk which is transient would be that performed by a (non-flying) drunk with a time machine -- although not as viewed by the drunk, but as viewed by an observer who is fixed at one point in space-time. (The model I have of space-time in my head for this problem is something like that which one sees in certain movies, where if you're not careful you can run into a copy of yourself.)
(Incidentally, I almost wrote "Michiko Kauktani" there; she's a literary critic for the New York Times. It turns out that Michiko is Shizuo's daughter, which I didn't know.)