In the 60’s, I. Gel’fand introduced the concept of generalized stochastic processes (Ch. III, Vol. 4 of his work on Generalized functions). For a generalized stochastic process Φ, he defines the concepts of stationarity (Φ(φ) and Φ(φ(⋅−t0)) have the same law) and of independence at every point (the random variable Φ(φ1) and Φ(φ2) are independent if φ1 and φ2 have disjoint supports).
Gel’fand especially introduces the complete class of Lévy white noises as generalized stochastic processes with characteristic functional of the form
with f a function that has a L\’evy-Khintchine representation.
Obviously, white noises are not the only stationary and independent at every point processes (ex: the weak derivative of a white noise). I am interested by a characterization of stationary and independent at every point processes. Especially, Gel’fand conjectured the following result.
If a generalized stochastic process Φ is stationary and independent at every point, then the characteristic function L of Φ has the form
with f a continuous function from Rn+1 to C with f(0)=0.
Is that result true? In order to express a kind of reciprocal result, can we characterize the functions f such that the previous functional is a characteristic functional (main problem: its positive-definiteness)?
Some people are extensively studied the positive-definiteness of functionals but wasn’t able to find references that are clearly answering my question.
Thanks for your attention.
I would suggest the reference , where J.N. Pandey addresses this problem.
 J.N. Pandey, “On the positive definiteness of a functional”, Canad. Math. Bull., Vol 22 (2), 1979