Much of ergodic theory deals with deterministic dynamical systems, whereas this talk deals with stochastic systems. Ergodic systems are insensitive to initial conditions, in the sense that statistical properties approach unique stationary values as time goes by. Here, all possible states of the system are mutually accessible, and no event or decision has long-lasting effects. Their treatment is mathematically much simpler than that of non-ergodic systems because much can be computed without specifying any dynamics. In reality, unfortunately, we are often faced with non-ergodic systems. Decisions cannot be reversed, missed opportunities are forever missed. Non-ergodic systems require a more careful mathematical treatment. We must distinguish between expectation values, averaged over all possible outcomes of a stochastic process, and time-averages, computed from individual realizations. An example closely related to the financial crisis will be discussed.
The first part of the talk consists of well-known (introductory) and less well-known (current research) examples. The second part of the talk is concerned with conceptual implications related to the concepts of time and randomness.