A theory of spatiotemporal random fields (S/TRF) is developed. More precisely, we are concerned with random fields whose properties are coordinated with the algebraic structure of the product set "Euclidean n-dimensional space R(n) x Time axis T." The Ito-Gel'fand concept of random distributions is extended in a suitable space-time context, on appropriate Schwartz spaces of functions. This extension allows us to construct a unified theory of S/TRF, which includes ordinary as well as generalized random fields. Particularly, we first introduce certain notions regarding the space-time structure assumed, and then we define a Hilbert space H-kappa of ordinary spatiotemporal random variables, where only the strong topology is considered. Such consideration leads to the space K of H-kappa-valued ordinary S/TRF (OS/TRF) X (s,t), (s,t) is-an-element-of R(n) x T. The space K is extended to the space G of generalized S/TRF (GS/TRF) X(q) which is established on Schwartz spaces Q = K and S in R(n) x T, and its properties associated to certain continuous linear functional-type representations are examined. The stochastic correlation structure of G is studied by means of the space-time covariance and structure functionals. These are continuous nonnegative-definite bilinear functionals belonging to the dual space Q' (= K' or S') of Q, and are linearly related to the ordinary correlation structure of K. Among the classes of S/TRF introduced, particularly attractive is the general class of random fields that are nonhomogeneous of order nu-in space direction and nonstationary of order mu-in time direction (S/TRF-nu/mu), on a properly defined "admissible" space Q-nu/mu subset-of Q. This considerably rich class is denoted by G-nu/mu and each element X(q) is-an-element-of G-nu/mu is the mapping X : Q-nu/mu --> H(G), where H(G) = L2(OMEGA, F, P) is the Hilbert space of random variables in space-time and (OMEGA, F, P) is the P-measurable probability space satisfying Kolmogorov's axioms. The set of all OS/TRF-nu/mu associated to a particular GS/TRF-nu/mu X(q) constitute a generalized representation set (GRS-nu/mu). The general form of the GRS-nu/mu representations is derived, and a decomposition of an OS/TRF-nu/mu in terms of an infinitely differentiable OS/TRF-nu/mu and a space homogeneous/time stationary S/TRF is established. It is shown that representations of GRS-nu/mu emerge in the context of stochastic partial differential operators, or polynomial space-time series with random coefficients in H-kappa. The elements of the space G-nu/mu have a complex space nonhomogeneous/time nonstationary stochastic correlation structure that, however, can be decomposed to a space homogeneous/time stationary part termed generalized spatiotemporal covariance of order nu/mu (GS/TC-nu/mu), and a space-time polynomial part. The GS/TC-nu/mu is a conditionally nonnegative-definite function in R(n) x T, to which a unique spectral representation is associated. One of the interesting feature of the space-time correlation decomposition is that stochastic inferences can be made and optimal linear Wiener-Kolmogorov estimators of discretely-valued OS/TRF-nu/mu can be derived solely in terms of GS/TC-nu/mu. These classes of S/TRF have important stochastic data processing applications in various areas of applied sciences, including environmental engineering, geohydrology and meteorology.
