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)