The theory of stochastic or random processes is the application of probability theory to situations where the random objects are functions. If one focuses on quantities that can be observed about a function, such as its pointwise values, its integrals against given test functions, its extreme values, and so on, a stochastic process is just an uncountable collection of random variables satisfying consistency conditions coming from the fact that the random variables are all observations of a function rather than a disparate collection of variables.
Random processes are a broad topic in both pure mathematical research and applications. This page is about random processes as a means of modelling in engineering and scientific applications. As such, most often the random function is a function of continuous time, time ordering and continuity play an important role, and one considers dynamical equations with random coefficients and/or driven by noise. These stochastic differential equations are the primary “consistency conditions” linking all the random variables associated with a stochastic process.Also, and especially in climate models, one may consider random fields which are a function of spatial coordinates, not time, or evolving random fields where the random function is a function of both time and space coordinates, leading to stochastic partial differential equations.
A stochastic (random) process consists of a family of random variables on common underlying probability space. If the variables form a sequence, then it is a discrete time process; if they are indexed by an interval of real numbers, it is a continuous time process.
A probability space consists of the following data:
The sample space S, which is the set of possible outcomes (of an experiment)
The event algebra A, where each event consists of a set of outcomes in S, and the collection of events constitutes a $\sigma$-algebra – it is closed under countable sequences of union, intersection and complement operations (and also set differences). Implied here is the fact that the empty set and whole sample space are events in A.
A measure function P, which assigns a probability to each event in A. P must be additive on countable disjoint unions, and must assign 1 to the whole event algebra A.
A random variable is a function $X$ from the sample space S into a range space V, which is measurable, which means: there is a $\sigma$-algebra of subsets of V, and the inverse image of every such subset under the function $X$ is an event in A.
A computer model of a random process needs a random number generator. Certain random processes are solutions of stochastic differential equations and can therefore be simulated by numerical solvers of these equations.
Terence Tao? has written about measure theory on abstract spaces and random processes, they are part of these online lecture notes:
Terence Tao: real analysis
Terence Tao: random matrices