Let $\mathscr{C}:=\{\gamma : \mathbb{R}\_+\rightarrow\mathbb{R}^n \mid \gamma \ \text{ continuous}\}$ be the set of all $\mathbb{R}^n$-valued paths over $[0,\infty)$. Endow $\mathscr{C}$ with the $\sigma$-algebra $\mathfrak{C}$ generated by all projections $\gamma \mapsto \gamma\_t$ (for $t\geq 0$ fixed). Let further


$$\phi : \mathscr{C}\rightarrow\mathscr{C} \ \ \text{ be given by } \ \phi(\gamma):=(\gamma\_{t+1})\_{t\geq 0} \ \text{ (left-shift),}$$ and call *$\phi$-invariant* any event $S\in\mathfrak{C}$ with $\phi^{-1}(S)=S$.


Do you know of examples for $\mathscr{C}$-valued stochastic processes $Y$ that attain $\phi$-invariant events with trivial probability, i.e. are such that $\mathbb{P}\_Y(S)\in\{0,1\}$ whenever $S\in\mathfrak{C}$ is $\phi$-invariant?



Shift-ergodic stochastic processes in continuous time
Do you not also want that $\mathbb{P}\_Y$ is $\phi$-invariant?


In any case, yes there are extremely many continuous-time continuous-path real-valued stochastic processes whose law is ergodic under the time-$1$-shift map. Of course, the most trivial example would be where
$$ \mathbb{P}(Y\_t=c \ \ \forall t \geq 0) = 1 $$
for some $c \in \mathbb{R}$. An almost-as-trivial case would be
$$ Y\_t = p(2\pi ft + \Phi) $$
where $p(\cdot)$ is a $1$-periodic continuous function, $f$ is an irrational number, and $\Phi$ is a random variable with $\Phi \sim \mathrm{Uniform}(0,2\pi)$. (I don't know if this would count as a "non-trivial example class" under your bounty statement!)


Perhaps the simplest "non-trivial" example would be the *Ornstein-Uhlenbeck* process:


(I hope there are no typos in the following!) Fix two parameters $\theta,\sigma>0$. Starting off with a two-sided Wiener process $(W\_t)\_{t \in \mathbb{R}}$ (meaning that $(W\_t)\_{t \geq 0}$ and $(W\_t)\_{t \leq 0}$ are independent Wiener processes), we can define a stochastic process $(Y\_t)\_{t \in \mathbb{R}}$ where for almost every sample path of the Wiener process, for all $t \in \mathbb{R}$,
\begin{align\*}
Y\_t &= \sigma \! \int\_{-\infty}^t e^{\theta(s-t)} \, dW\_s \quad\quad\quad\quad\quad\quad\ \, \text{(Riemann-Stieltjes integral)} \\
&= \sigma \left( W\_t - \theta \! \int\_{-\infty}^t e^{\theta(s-t)} W\_s \, ds \right) \quad \text{(classical integral, obtained by integration by parts)}.
\end{align\*}
This is the unique two-sided-time strong solution of the stochastic differential equation
$$ dY\_t = -\theta Y\_t \, dt + \sigma dW\_t, $$
or equivalently (as a random differential equation)
$$ \frac{d(Y\_t - \sigma W\_t)}{dt} = -\theta Y\_t. $$
$\text{[}$The one-sided-time solution $(Y\_t^{(y\_0)})\_{t \geq 0}$ of this equation starting at any $y\_0 \in \mathbb{R}$ would be
\begin{align\*}
Y\_t^{(y\_0)} &= e^{-\theta t} \left( y\_0 + \sigma \! \int\_0^t e^{\theta s} \, dW\_s \right) \\
&= \sigma W\_t + e^{-\theta t} \left( y\_0 - \sigma\theta \! \int\_0^t e^{\theta s} W\_s \, ds \right). \text{]}
\end{align\*}
A one-sided-time stochastic process with the law of $(Y\_t)\_{t \geq 0}$ (or a two-sided-time stochastic process with the law of $(Y\_t)\_{t \in \mathbb{R}}$) is called an Ornstein-Uhlenbeck process.


*Sketch-proof that OU process is shift-ergodic.* Firstly, working in two-sided-time,


* one can check that replacing $(W\_t)\_{t \in \mathbb{R}}$ with $(W\_{t+1}-W\_1)\_{t \in \mathbb{R}}$ transforms $(Y\_t)\_{t \in \mathbb{R}}$ to $(Y\_{t+1})\_{t \in \mathbb{R}}$;
* since the Wiener process has memoryless stationary increments, one can show that the law of $(W\_t)\_{t \in \mathbb{R}}$ is mixing and hence ergodic with respect to the map $(\gamma\_t) \mapsto (\gamma\_{t+1}-\gamma\_1)$;


and so the law of $(Y\_t)\_{t \in \mathbb{R}}$ is ergodic with respect to the map $(\gamma\_t) \mapsto (\gamma\_{t+1})$. Finally, having ergodicity in two-sided time guarantees ergodicity in the one-sided-time restriction.




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In general, given any $\phi$-invariant probability measure $\mathbb{P}$ on $\mathscr{C}$, one can express $\mathbb{P}$ as a weighted average of ergodic $\phi$-invariant probability measures on $\mathscr{C}$. This is essentially the *ergodic decomposition theorem*, which holds for any Borel-measurable dynamical system on a Polish space; in this case, the $\sigma$-algebra $\mathfrak{C}$ is precisely the Borel $\sigma$-algebra of the topology of uniform convergence on compact sets, which is a Polish topology.


