The fact that it's biased is irrelevant. The question whether it will get close to any point is quite complicated, though with it being essentially a random wal

The fact that it's biased is irrelevant. The question whether it will get close to any point is quite complicated, though with it being essentially a random walk with drift the chance that will get above its initial point is probably less than 1. This's got nothing to do with ergodic theory though, but feel free to use that word in whatever weird distorted way you like, I literally can't seem to stop you.