ABSTRACT
This chapter analyses time-dependent stochastic systems. A time-dependent function of a random variable is referred to as a random process. For example, the height of a burning candle can be thought of as a process that progresses with time. Real clocks, consisting of material that are made of atomic- and subatomic-scale particles that execute random motions, of course, progress only forward in time. The scent molecules are much more randomly distributed after they leave the bottle than they were in the bottle. A quantum mechanical oscillator, of course, has some probabilistic aspects. However, its motion is independent of the direction of time. The effect of the spring on the mass M is modeled by a discretized time Lagrangian sum. The Lagrangian in classical mechanics is equal to the difference between the kinetic and potential energies.
