I am following the recent paper from Mourat and Weber (GLOBAL WELL-POSEDNESS OF THE DYNAMIC $\Phi^4_3$ MODEL ON THE TORUS) where we find the following:
Where the strange $i$-like symbol is defined as:
and $\xi$ is the space time white noise that we constructed as:
I couldn't derive relation (4.1)
Attempt:
The Fourier transform of a distribution is an extension of the ordinary Fourier transform for $L^1$ functions: $$ \hat i(t,\omega) = \int i(t)(x) e^{-2\pi i w\cdot x} \, dx $$
but $i(t)$ is a distribution, therefore
$$ \hat i(t,\omega) = i(t)( \psi_\omega) $$
where $\psi_\omega(x) = e^{-2\pi i w\cdot x}$ now, since $i(t)$ has an integral form
$$\hat i(t,\omega) = \big(\int_{-\infty}^t P_{t-s}(\xi(s))\, ds\big)( \psi_\omega) \\ =\int_{-\infty}^t P_{t-s}(\xi(s)( \psi_\omega))\, ds $$
but $$\xi(s)( \psi_{\omega}) = \sum_{\omega'} \int_{-\infty}^\infty \hat{\psi_{\omega}} (t,\omega') dW(t,\omega') = ? $$
How should we proceed to obtain (4.1)?
trying a bit further:
Since $(\psi_\omega)_{\omega \in \Bbb{Z}^d}$ is an orthonomal set in the Thorus, one might reasonably write:
$$\xi(s)( \psi_{\omega}) = \sum_{\omega'} \int \hat{\psi_{\omega}} (t,\omega') dW(t,\omega') = \int_{-\infty}^\infty dW(t,\omega) =? $$
It may make more sense to write
$$\hat i(t,\omega) = \big(\int_{-\infty}^t P_{t-s}(\xi(s))\, ds\big)( \psi_\omega) \\ =\int_{s= -\infty}^t P_{t-s}(\xi(s))( \psi_\omega)\, ds\\ =\int_{s = -\infty}^t \sum_{\omega'} \int_{t = -\infty}^\infty \hat{P_{t-s}\psi_{\omega}} (t,\omega') dW(t,\omega') ds \overset{?}{=}\int_{s = -\infty}^t \hat{P}_{t-s}(\omega) dW(s,\omega)\\ $$