Let $\Omega$ be a Polish locally compact space and $(\Omega, \mathscr{F}, \mathbb{P})$ be a probability space. Consider a measurable map \begin{align*} \theta\colon T\times \Omega &\to \Omega\\ (t,\omega)&\mapsto \theta_{t}\omega \end{align*} such that $\theta_{t+s}=\theta_{t}\circ \theta_{s}$ for every $s, t\in T$ and $\theta_{t}$ preserves $\mathbb{P}$ for every $t\in T$, that is, $\theta_{t}\mathbb{P}=\mathbb{P}$ for every $t\in T$. In other words, $(\theta_{t})$ is a measure preserving dynamical system on $(\Omega, \mathscr{F}, \mathbb{P})$.

**Definition 1 (Random Dynamical Systems and Invariant Measures)**:
Let $(\Omega, \mathscr{F}, \mathbb{P})$ and $(\theta_{t})$ as above and consider a compact metric space $X$ (endowed with its Borel $\sigma$-algebra $\mathscr{B}$).

A measurable map
\begin{align*}
\varphi\colon T\times \Omega\times X &\to X\\
(t,\omega,x)& \mapsto \varphi(t,\omega)x
\end{align*}
is called a *random dynamical system* on $X$ over the measure preserving dynamical system $(\theta_{t})$ if

a) $x\mapsto \varphi(t,\omega)x$ is continuous for $\mathbb{P}$-almost every $\omega$ and every $t\in T$.

b) $\varphi(t+s, \omega)=\varphi(t,\theta_{s}\omega)\circ \varphi(s,\omega)$ for all $t,s\in T$, and $\varphi(0,\omega)=id$, for $\mathbb{P}$-almost every $\omega$.

A random dynamical system $\varphi$ induces a dynamical system over $\Omega\times X$ given by
$$
\Theta(t)(\omega, x)=(\theta(t)\omega, \varphi(t,\omega)x),\quad t\in {T}
$$
called the *skew product* associated with $\,\varphi$.

A probability measure $\mu$ on $\Omega\times X$ is said to be a $\varphi$-*invariant measure* for the random dynamical system $\varphi$, if it satisfies

$\Theta(t)\mu=\mu$ for all $t\in T$;

$\Pi\mu=\mathbb{P}$, where $\Pi$ is the natural projection on the first factor defined by $\Pi(\omega,x)=\omega$.

Let $\mu$ be a $\varphi$-invariant measure. Since $\Pi\mu=\mathbb{P}$, we can consider the disintegration $d\mu=\mu_{\omega}\, d\mathbb{P}(\omega)$ of $\mu$ with respect to $\mathbb{P}$. A necessary and sufficient condition for $\mu$ to be a $\varphi$-invariant measure is that for every $t\in T$ \begin{equation}\label{inva} \varphi(t,\omega)\mu_{\omega}=\mu_{\theta_{t}\omega} \end{equation} for $\mathbb{P}$-almost every $\omega$.

**Definiton 2:** We say that $\mu$ is a **random Dirac measure** if
there is measurable map $\Phi\colon \Omega\to X$ such that
$$
\mu_{\omega}=\delta_{\Phi(\omega)}
$$
for $\mathbb{P}$-almost every $\omega$.

**MY QUESTION:** I am looking for examples of Random Dynamical Systems (References of Papers) in which there is a finite($\geq$ 2) number of ergodic $\varphi$-invariant measures,
which are, additionally, random Dirac measures.

I am most interested in the continuous time cases, but examples (References of Papers)in the discrete time setting is also welcome.