  We study the interplay between the regularity of paths and Hamiltonians in
the theory of pathwise Hamilton-Jacobi equations with the use of interpolation
methods. The regularity of the paths is measured with respect to Sobolev,
Besov, H\"older, and variation norms, and criteria for the Hamiltonians are
presented in terms of both regularity and structure. We also explore various
properties of functions that are representable as the difference of convex
functions, the largest space of Hamiltonians for which the equation is
well-posed for all continuous paths. Finally, we discuss some open problems and
conjectures.
