# Abstract characterization of white noise

Let $$\xi$$ be a random variable valued in the space of Schwartz distributions $$\mathcal{S}'(\mathbb{R}^d)$$.

For any open set $$R\subset\mathbb{R}^d$$ let $$\Sigma(R)$$ be the $$\sigma$$-algebra generated by $$\{\xi(f)\,|\,f\in\mathcal{S}(\mathbb{R}^d), \,\textrm{supp}\,f\subset R\}$$.

For any $$\lambda\in(0,\infty)$$ the random variable $$\xi_\lambda$$ valued in $$\mathcal{S}'(\mathbb{R}^d)$$ is defined by $$\xi_\lambda(f) = \xi(f_\lambda)$$ for all $$f\in\mathcal{S}(\mathbb{R}^d)$$, where $$f_\lambda(x) = \lambda^{d/2}\,f(\lambda x)$$.

Assume that:

(1) the $$\sigma$$-algebras $$\Sigma(R_1)$$ and $$\Sigma(R_2)$$ are independent for any disjoint open sets $$R_1,R_2\subset\mathbb{R}^d$$.

(2) for any $$\lambda\in(0,\infty)$$ the law of $$\xi_\lambda$$ coincides with the law of $$\xi$$.

Is it true that under the above assumptions $$\xi$$ must be the white noise?

• What is your definition of the white noise? Oct 6 at 16:03
• Given the adopted point of view is that of random distributions rather than generalized stochastic processes, white noise is the Borel probability measure on the space of distributions whose characteristic function is $f\mapsto\exp(-\frac{1}{2}\|f\|_{L^2}^2)$, for arbitrary Schwartz function $f$. Oct 6 at 17:27
• possibly useful reference sma.epfl.ch/~rdalang/articles/AOP1168.pdf Oct 6 at 17:36

These assumptions are not sufficient. Take $$d=1$$ and for simplicity let us work with the antiderivative of $$\xi$$. You are then asking if a process with independent increments and Brownian scaling is necessarily a Brownian motion. Any $$\alpha$$-stable Levy process $$L$$ has independent increments and scaling $$(L_t)_{t\geq 0}\sim\lambda^{1/\alpha}(L_{\lambda t})_{t\geq0}$$, so the process $$X_t=L_{t^{\alpha/2}}$$ is a counterexample.
My guess is that translation adding translation invariance and nontriviality (note that actually $$\xi\equiv 0$$ is also a counterexample) to the assumptions should be sufficient.
• Thanks. So the question is whether $\xi$ satisfying (1), (2) and (3) translational invariance is necessarily proportional to the white noise. Oct 7 at 8:18