It is well-known that one can construct a Hilbert space of zero mean, finite variance random variables such that every random variable of this type $X$ can be represented as a linear combination of a sequence of random variables $F_i$ with respect to probability:
$$X = \sum_{i=1}^\infty \frac{\text{cov}(X,F_i)}{\text{var}(F_i)}F_i$$
Is there any such meaningful Hilbert space construction for stochastic processes $X(t)$?
Yes, any square-integrable process on a finite interval $[a,b]$ can be written $X_t=\sum_i Z_i e_i(t)$ where $Z_i$ are uncorrelated, $e_i$ are an orthonormal basis of $L^2$, and the convergence is in $L^2$. This is called the Karhunen-Loeve decomposition, and much more information can be found on the Wiki page: https://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem