  We give an intrinsic meaning to the Hamilton-Jacobi equation arising from
mean-field spin glass models in the viscosity sense, and establish the
corresponding well-posedness. Originally defined on the set of monotone
probability measures, these equations can be interpreted, via an isometry, to
be defined on an infinite-dimensional closed convex cone with an empty interior
in a Hilbert space. We prove the comparison principle, and the convergence of
finite-dimensional approximations furnishing the existence of solutions. Under
additional convexity conditions, we show that the solution can be represented
by a version of the Hopf-Lax formula, or the Hopf formula on cones. Previously,
two notions of solutions were considered, one defined directly as the Hopf-Lax
formula, and another as limits of finite-dimensional approximations. They have
been proven to describe the limit free energy in a wide class of mean-field
spin glass models. This work shows that these two kinds of solutions are
viscosity solutions.
