I have a question regarding ergodicity in infinite dimensional spaces.

Let $\mathcal{D}$ be the space of distributions on a Schwartz space, and let $\mu$ be the white noise process which exists by the Bochner-Minlos theorem.

We can define the translation $\tau_x \phi$ of a distribution $\phi$ by: $$ \langle \tau_x \phi, \varphi \rangle = \langle \phi, \tau_x \varphi \rangle, $$ for all $\varphi \in C^\infty_c(\mathbb{R}^n)$, where $\tau_x \varphi$ is the usual translation of a function. The set of translations form a group acting on $\mathcal{D}$.

My first question is:

**Is the group $\lbrace \tau_x : x\in \mathbb{R}^n \rbrace$ ergodic with respect to the white noise measure?**

Now let $\varphi \in C^\infty_c(\mathbb{R}^n)$, then we can define the convolution $\varphi * \zeta(\omega)$, where $\zeta$ is a white-noise distributed variable. This is a $C^\infty$ random variable. My second question is:

**Is $\varphi * \zeta(\omega)$ ergodic with respect to translations?**

My intuition is that the first answer is true, and the second answer is "true, assuming the support is sufficiently small", but I don't even know what tools to use to tackle this problem.