“Process” always makes me confused…

So I prefer using the mathematically equivalent word “Field”:

$\boxed{\text{A random field is a random function, not a single function.}}$

Or:

$\boxed{\text{A random field is a mapping from outcomes to functions.}}$

That is,

$X:\Omega \to S$

where $S$ means “the set of all possible functions (with their index set)”

If the index set is finitely counted:

$\boxed{\text{A random field is a collection of random variables with a joint probability structure.}}$

• For Gaussian fields: $\text{mean}+\text{covariance}\quad\Longrightarrow\quad\text{full law}.$
• For general random fields: $\text{mean}+\text{covariance}\quad\not\Longrightarrow\quad\text{full law}.$

So the empirical approach is to build a statistical signature of the field:

• Common: Marginals, covariance, copulas, tail dependence, spectra
• Not common: basis coefficient laws, level-set geometry…

No finite experiment fully recovers the infinite-dimensional law, but with many samples you can approximate its behavior at the spatial, temporal, amplitude, and geometric resolutions relevant to your problem.