One can define (continuous) Gaussian free field as follows: one can consider some orthonormal basis $(\psi_k)_{k=1}^{\infty}$ in the Sobolev space $H^1(\Omega)$ (here $\Omega \subset \mathbb{R}^d$) and then consider $\sum_{k=1}^{\infty} \xi_k \psi_k$ where $\xi_k$ are independent, identically distributed (normalized) normal random variables. When $d=1$ this sum makes sense as a continuous function: however when $d>1$ this random sum only makes sense as a distribution. I wonder what will be the difference if we consider $L^2(\Omega)$ instead of $H^1(\Omega)$ (for example $L^2(\mathbb{R}^d)$ with Hermite polynomials as a basis). Is this object well defined/was it studied? I will welcome any references
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$\begingroup$ In general, a sum of the form $\sum \xi_k \psi_k$, where $\psi_k$ is an orthonormal set in a Hilbert space $H$, will diverge almost surely in the topology of $H$. It is always possible to find some larger space $W$ with a weaker topology (possibly again Hilbert or Banach), with a dense embedding of $H$ into $W$, such that the sum converges i.p. in $W$. The resulting "object" depends entirely on the choice of $W$, though. $\endgroup$– Nate EldredgeCommented Jan 17, 2020 at 19:29
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$\begingroup$ To have some receptacle which is a bit more canonical, one can also take $W=\mathcal{S}'(\mathbb{R}^d)$ for $\Omega=\mathbb{R}^d$. $\endgroup$– Abdelmalek AbdesselamCommented Jan 17, 2020 at 19:45
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3$\begingroup$ Also, the case of $L^2$ with the $\psi$'s given by Hermite functions leads to white noise rather than the GFF. $\endgroup$– Abdelmalek AbdesselamCommented Jan 17, 2020 at 19:47
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$\begingroup$ @AbdelmalekAbdesselam thank you for your comment-could you please give me some reference for this statement regarding white noise? $\endgroup$– truebaranCommented Jan 18, 2020 at 12:47
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$\begingroup$ I didn't use a reference. It's easy to figure out by hand. $\endgroup$– Abdelmalek AbdesselamCommented Jan 20, 2020 at 16:30
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