Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. Let $(W(t))_{x \in \mathbb{R}^d}$ be a Gaussian random field. Then, we can define Wiener integral $\int_{\mathbb{R}^d} f(\xi) \, dW(\xi)$ for any $f \in L^2(\mathbb{R}^d)$. For the definition of the Wiener integral, see the following question: https://math.stackexchange.com/questions/4201253/gaussian-random-fields-and-wiener-integral
For $f \in L^2(\mathbb{R}^d)$ and $\varphi:\mathbb{R}^d \to \mathbb{R}$, define $Y_{f,\varphi}$ by $$ Y_{f,\varphi}:= \int_{\mathbb{R}^d} e^{i\varphi(\xi)} f(\xi) \, dW(\xi). $$ My question is whether the distribution of $Y_{f,\varphi}$ is independent of $\varphi$ or not. And if the answer is yes, how to prove it?
I explain the background of this question. I'm reading the following paper:
COLLOT, CHARLES; DE SUZZONI, ANNE-SOPHIE Stability of steady states for Hartree and Schrödinger equations for infinitely many particles. https://ahl.centre-mersenne.org/item/AHL_2022__5__429_0/
This paper define $Y$ by $Y=Y_{f,\varphi}$ where $\varphi(\xi) = -it(m+|\xi|^2) + ix\xi$, and claim that the distribution of $Y$ is invariant under the space and time translations. However, I cannot prove it.
