  An operator satisfies the Global Comparison Property if anytime a function
touches another from above at some point, then the operator preserves the
ordering at the point of contact. This is characteristic of degenerate elliptic
operators, including nonlocal and nonlinear ones. In previous work, the authors
considered such operators in Riemannian manifolds and proved they can be
represented by a min-max formula in terms of L\'evy operators. In this note we
revisit this theory in the context of Euclidean space. With the intricacies of
the general Riemannian setting gone, the ideas behind the original proof of the
min-max representation become clearer. Moreover, we prove new results regarding
operators that commute with translations or which otherwise enjoy some spatial
regularity.
