Let E′ be the dual of a nuclear Fréchet space E and L*(t) the adjoint operator of a diffusion operator L(t) of infinitely many variables, which has a formal expression:
A weak form of the stochastic differential equation
dX(t) = dW(t) + L*(t)X(t)dt
is introduced and the existence of a unique solution is established. The solution process is a random linear functional (in the sense of I. E. Segal) on a space of generalized functionals on E′. The above is an appropriate model for the central limit theorem for an interacting system of spatially extended neurons. Applications to the latter problem are discussed.