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?