Stochastic process, generalized

A stochastic process $X$ depending on a continuous (time) argument $t$ and such that its values at fixed moments of time do not, in general, exist, but the process has only "smoothed values" $X ( \phi )$ describing the results of measuring its values by means of all possible linear measuring devices with sufficiently smooth weight function (or impulse transition function) $\phi ( t)$. A generalized stochastic process $x ( \phi )$ is a continuous linear mapping of the space $D$ of infinitely-differentiable functions $\phi$ of compact support (or any other space of test functions used in the theory of generalized functions) into the space $L _ {0}$ of random variables $X$ defined on some probability space. Its realizations $x ( \phi )$ are ordinary generalized functions of the argument $t$. Ordinary stochastic processes $X ( t)$ can also be regarded as generalized stochastic processes, for which

$$X ( \phi ) = \int\limits _ {- \infty } ^ \infty \phi ( t) X ( t) d t ;$$

this is particularly useful in combination with the fact that a generalized stochastic process $X$ always has derivatives $X ^ {(n)}$ of any order $n$, given by

$$X ^ {(n)} ( \phi ) = ( - 1 ) ^ {n} X ( \phi ^ {(n)} )$$

(see, for example, Stochastic process with stationary increments). The most important example of a generalized stochastic process of non-classical type is that of white noise. A generalization of the concept of a generalized stochastic process is that of a generalized random field.

For references, see Random field, generalized.

References

[a1]	I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1964) (Translated from Russian)