When it occurs in one dim
When it occurs in one dimension with equal probabilities it is an example of a martingale. According to Polya's theorem, it will never return to the origin if it occurs in greater than two dimensions, and if it occurs on a lattice, the probability of reaching any point is one as time goes to infinity. The simple type is an example of a Markov chain, and in both one and two dimensions, the root mean square displacement after n iterations is proportional to the square root of n. The mathematical basis for such processes as diffusion and Brownian motion, FTP, identify this process in which every iteration consists of taking a step in some direction with some probability, often illustrated with a drunken sailor.
When it occurs in one dimension with equal probabilities it is an example of a martingale.
random walk