What could be an example of 'time' in a stochastic process where 'time' is not a natural time? I need any example of a stochastic process where 'time' is not a natural/real-world element what we generally call the time, and measure it using clocks/watches. What could be an example of 'time' in a stochastic process where 'time' is not a time?

One example is a Poisson point process in (say) the plane, that associates to each measuable set $A\subseteq \mathbb R^2$ a Poisson random variable $N(A)$ with expectation $\lambda(A)$ (where $\lambda$ is some measure), such that if $\lambda(A\cap B)=0$ then $N(A)$ and $N(B)$ are independent and $N(A\cup B)=N(A)+N(B)$. Here the ''time'' space is something like $\mathcal B(\mathbb R^2)$, an object far larger (or at least more complicated) than $\mathbb R$.