In this book we will study dynamical systems driven by noise. Noise is something that changes randomly with time, and quantities that do this are called stochastic processes. When a dynamical system is driven by a stochastic process, its motion too has a random component, and the variables that describe it are therefore also stochastic processes. To describe noisy systems requires combining differential equations with probability theory. We begin, therefore, by reviewing what we will need to know about probability.

Random variables and mutually exclusive events

Probability theory is used to describe a situation in which we do not know the precise value of a variable, but may have an idea of the relative likelihood that it will have one of a number of possible values. Let us call the unknown quantity X. This quantity is referred to as a random variable. If X is the value that we will get when we roll a six-sided die, then the possible values of X are 1, 2, …, 6. We describe the likelihood that X will have one of these values, say 3, by a number between 0 and 1, called the probability. If the probability that X = 3 is unity, then this means we will always get 3 when we roll the die. If this probability is zero, then we will never get the value 3.