# What are all the stationary and pointwise independent random processes?

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 $\Phi$, he defines the concepts of stationarity ($\Phi(\varphi)$ and $\Phi(\varphi(\cdot - t_0))$ have the same law) and of independence at every point (the random variable $\Phi(\varphi_1)$ and $\Phi(\varphi_2)$ are independent if $\varphi_1$ and $\varphi_2$ 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.

Conjecture:
If a generalized stochastic process $\Phi$ is stationary and independent at every point, then the characteristic function $L$ of $\Phi$ has the form

with $f$ a continuous function from $\mathbb{R}^{n+1}$ to $\mathbb{C}$ 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.