Find the characteristic functional of a stochastic process

Question

This doubt arose quite unexpectedly when I was asking another question. I explain:

Let $X = (X_t)_{t \in \mathbb Z}$ and $Y = (Y_t)_{t \in \mathbb Z}$ be two independent and strictly stationary stochastic processes with real values and mean zero, i.e. $E[X_t]=E[Y_t]=0,\quad \forall\, t \in \mathbb Z$.

Now we will define two stochastic process:

1. Consider $U\sim \hbox{Bernoulli(p)}$ a random variable independent of $X$ and $Y$. Define $Z = (Z_t)_{t \in \mathbb Z}$ as: \begin{equation}\label{abc}\tag{I} Z=\begin{cases} X, & \text{ if $U =1$}\\ Y, & \text{ if $U =0$} \end{cases} \end{equation} Note that I can writte $Z= X 1_{[U=1]} + Y 1_{[U=0]}$. Moreover, according to same question cited above, $Z$ is strictly stationary.
2. Consider $U=(U_t)_{t \in \mathbb Z}$ iid sequence of random variables such that $U_t \sim \hbox{Bernoulli(p)}$. Moreover, $U$ is independent of $X$ and $Y$. Define $W = (W_t)_{t \in \mathbb Z}$ as: \begin{equation}\label{abcd}\tag{II} W_t=\begin{cases} X_t, & \text{ if $U_t =1$}\\ Y_t, & \text{ if $U_t =0$} \end{cases} \end{equation} In this case, I can writte $W_t= X_t 1_{[U_t=1]} + Y_t 1_{[U_t=0]}$ for all $t$. I believe that $W$ is also strictly stationary.

Before, according to this topic and its answer, we can treat stochastic processes as random maps with values in the space of functions and compute their characteristic functional in the same way that we compute the characteristic function of random variables.

I will calculate the ch. functional of $Z$. Let $q=1-p$ and $\lambda: \mathbb Z \to \mathbb R$ a real function: $$ \begin{aligned} \varphi_Z(\lambda) &= E[ e^{i \langle \lambda, Z\rangle }] = E[ e^{i \langle \lambda, Z\rangle } (1_{[U=1]} + 1_{[U=0]})] = E[ e^{i \langle \lambda, Z\rangle } 1_{[U=1]}] + E[ e^{i \langle \lambda, Z\rangle } 1_{[U=0]}]\\ &=E[ e^{i \langle \lambda, X\rangle } 1_{[U=1]}] + E[ e^{i \langle \lambda, Y\rangle } 1_{[U=0]}]\\ &=E[ e^{i \langle \lambda, X\rangle }]p + E[ e^{i \langle \lambda, Y\rangle }]q\\ &=\varphi_X(\lambda)p + \varphi_Y(\lambda)q\\ \end{aligned} $$

For me, the second case is more complicated to find a closed formula for the characteristic functional as in the case of $Z$. The most I can do is to compute the characteristic functions of the marginals(as the same way of $Z$): $$\varphi_{W_t}(s) = \varphi_{X_t}(s)p + \varphi_{Y_t}(s)q, \quad s \in \mathbb R$$ But I don't know if it helps to build the characteristic functional of $W$, also using strict stationarity. Is there a closed form or is it simply not worth thinking about?