A stochastic process such as a random walk can take on different "realizations" (where each realization has a particular initial condition and a particular sequence of steps). A stochastic process is said to be ergodic if its average behavior over one (sufficiently long) realization is equivalent to its average behavior over all possible realizations. A key reason to consider whether a process is ergodic is to determine if, given enough time, the process passes through all possible states with equal probability. If it so, the ultimate behavior of the process is not dependent on the initial conditions.
- Statistical Physics, Dynamical Systems