Showing changes from revision #8 to #9:
Added | Removed | Changed
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 funcion, 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 applications and sciences, scientific especially 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. Thesestochastic differential equations are the primary “consistency conditions” linking all the random variables associated with a stochastic process.Also, and especially in climate models . , As one such, may most consider often random fields which are a function of spatial coordinates, not time, or evolving random fields where the random function is a function of continuous both time, time ordering and continuity space play coordinates, an leading important to role, and one considers dynamical equations with random coefficients and/or driven by noise. These stochastic partial differential equation equations s . are the primary “consistency conditions” linking all the random variables associated with a stochastic process. One may also 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.
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