$\boxed{\text{A random process is a random trajectory, not a single trajectory.}}$

Or more formally:

$\boxed{\text{A random process is a function from outcomes to trajectories.}}$

That is,

$X:\Omega \to S^T$

where $S^T$ means “the set of all possible trajectories from time/index set $T$ into state space $S$.”

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

The random variables in a process may have different kinds of relationships.

For Gaussian fields:

$mean+covariance⟹full \ law$

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:

${marginals,covariance,increment laws,higher cumulants,copulas,tail dependence,spectra,level-set geometry,basis coefficient laws}.$

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.