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Under large mathematical conditions, the kowledge of the state probability density of a nonlinear stochastic distributed system is completely equivalent to the knowledge of all its state moments; and as a consequence, it may be interesting to investigate analysis techniques based on the study of these moments only. A method is herein proposed, which avoids the use of stochastic partial differential equations but rather defines the system by its infinitesimal transition moments. When the nonlinearities so involved by the system are polynomials with respect to the state, then the state moments satisfy an infinite set of linear differential integral equations. When such is not the case, then Galerkin's approximations are useful, and this approach is supported by functional continuity properties.