Let $X=(X_n,n\in \mathbb{N})$ be an arbitary $E$-valued stochastic process defined on probability space $\Omega$. Then how can we defined the shift operator?
Usually, $\Omega = E^\mathbb{N}$ and $X$ is taken to be the identity operator on $\Omega$ so that the shift operator can be defined by $\tau:\Omega \rightarrow \Omega$ where $(\omega_n,n\in \mathbb{N}) \mapsto (\omega_{n+1},n\in \mathbb{N})$ so that $X_n = X_0 \circ \tau^n$. However, how would we define $\tau$ if we had a general $\Omega$ and $X$?