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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 $$L(\varphi) =\exp\left( \int f(\varphi(t)) \mathrm{d}t \right),$$ 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 $$L(\varphi) =\exp\left( \int f(\varphi(t),\varphi^{(1)}(t),\cdots, \varphi^{(n)}(t)) \mathrm{d}t \right),$$ 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.

Thanks for your attention.

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I would suggest the reference [1], where J.N. Pandey addresses this problem.

[1] J.N. Pandey, ``On the positive definiteness of a functional'', Canad. Math. Bull., Vol 22 (2), 1979

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  • $\begingroup$ This note is clearly related with my second question (condition of positive-definiteness), but doesn't say anything on the conjecture. Indeed, Pandey is focusing on the case we already know that $L(\varphi)$ has the form expressed in my conjecture. He shows that a sufficient condition on $f$ expressed by Gel'fand is not necessary (with a counter-example) and gives himself a necessary (but not sufficient) condition. At least, it shows that one conjecture of Gel'fand was false! Thank you. $\endgroup$
    – Goulifet
    Mar 7 '14 at 16:02

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