Here, I am considering one of the simplest random dynamical systems that one can consider, and yet I realise that I do not know the answer to one of the most basic questions that one can ask about it!


Let $\Omega$ be the set of all continuous functions $\omega:\mathbb{R}\to\mathbb{R}$ satisfying $\omega(0)=0$. Let $\mathcal{F}$ be the smallest $\sigma$-algebra on $\Omega$ satisfying the requirement that the map $\omega \mapsto \omega(t)$ is $(\mathcal{F},\mathcal{B}(\mathbb{R}))$-measurable for all $t \in \mathbb{R}$. (It is known that $\mathcal{F}$ coincides with the Borel $\sigma$-algebra of the topology of uniform convergence on compact sets.)


For any $\omega \in \Omega$, define $\theta\omega \in \Omega$ by $\theta\omega(t)=\omega(t+1)-\omega(1)$ for all $t \in \mathbb{R}$. For each $\alpha \in \mathbb{R}$, let $R\_\alpha:\mathbb{S}^1 \to \mathbb{S}^1$ denote the anticlockwise rotation through angle $2\pi\alpha$. Define the function



$\begin{align}
\Theta \, : \, \Omega \times \mathbb{S}^1 \ &\to \ \Omega \times \mathbb{S}^1 \\
\Theta(\omega,x) \ &= \ (\theta\omega,R\_{\omega(1)}(x)).
\end{align}$




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> > Find the set of all $\Theta$-invariant probability measures $\mu$ on $(\Omega \times \mathbb{S}^1, \mathcal{F} \otimes \mathcal{B}(\mathbb{S}^1))$ with the property that
> >  $$ \mu(E \times \mathbb{S}^1) = \mathbb{P}\_W(E) \ \ \ \forall E \in \mathcal{F} $$
> >  where $\mathbb{P}\_W$ denotes the unique probability measure on $(\Omega,\mathcal{F})$ under which the stochastic processes $(\omega \mapsto \omega(t))\_{t \geq 0}$ and $(\omega \mapsto \omega(-t))\_{t \geq 0}$ are independent Wiener processes.
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One obvious measure with the desired properties is $\mathbb{P}\_W \otimes \lambda$, where $\lambda$ denotes the (normalised) Lebesgue measure on $\mathbb{S}^1$. But I don't know if there are any others.



Invariant and periodic measures of the random dynamical system on the circle generated by $d\theta_t=dW_t$
So I think there's a very direct argument to show that if $\mathbb P\_W\times\lambda$ is ergodic then it's the unique invariant measure projecting onto $\mathbb P\_W$. Let $\bar\Omega=\Omega\times S^1$ and define a rotation $R\_s\colon \bar\Omega\to\bar\Omega$ by $(\omega,t)\mapsto (\omega,s+t\bmod 1)$. Notice that if $\mu$ is an invariant measure, then so is $R\_s^\*\mu$. If $\mu$ is an invariant measure projecting to $\mathbb P\_W$, then $\int\_{S^1}R\_s^\*\mu=\mathbb P\_W\times\lambda$. Since $\mathbb P\_W\times \lambda$ is ergodic, you deduce $R\_s^\*\mu=\mathbb P\_W\times\lambda$ for almost every $s$. 


