Independence of stochastic processes Suppose that $(X_t)$ and $(Y_t)$ are stochastic processes defined on the same probability space whose sample paths belong to some Hilbert space $K$ (or more generally some function space). We may view these processes as $K$-valued random variables, hence we may talk about their independence as random variables.

Is the independence of stochastic processes $(X_t)$ and $(Y_t)$ equivalent to the independence of the corresponding $K$-valued random variables?

 A: The real question is: are $X = (X_t)$ and $Y = (Y_t)$ indeed $K$-valued random variables? This is not so obvious in continuous time, and in fact the answer may depend on the choice of the $\sigma$-field in $K$.
If $K$ is equipped with the product $\sigma$-field (generated by cylinder sets), then the answer is clearly "yes": $X = (X_t)$ and $Y = (Y_t)$ are $K$-valued random variables, and they are independent.
On the other hand, if $K$ is equipped with the Borel $\sigma$-field associated with the norm in $K$, then the answer can be "no". For example, take $\Omega = [0, 1]$ with the Lebesgue $\sigma$-field, consider a non-measurable set $E \subset \Omega$, and define the process $X_t$ so that $X_t(\omega) = 1$ if $t = \omega$ and $\omega \in E$, $X_t(\omega) = 0$ otherwise. Then $X_t$ has bounded, Borel-measurable paths, but $$\{\omega \in \Omega : \|X(\omega)\| < 1\} = E$$ is not measurable (here $\|X(\omega)\| = \sup \{|X_t(\omega)| : t \in [0, 1]\}$).
What happens if we know a priori that $X$ and $Y$ are measurable with respect to the Borel $\sigma$-field? I guess it is still possible to cook up an example in which $X$ and $Y$ may fail to be independent as $K$-valued variables, but I do not know that.
If, however, your processes are regular enough (right-continuous with left limits should do), then independence over the cylinder $\sigma$-field should imply independence over the Borel $\sigma$-field.
