The Parisi formula is a self-contained description of the infinite-volume limit of the free energy of mean-field spin glass models. We showthat this quantity can be recast as the solution of a Hamilton–Jacobi equation in the Wasserstein space of probability measures on the positive half-line.

In the paper we investigate the regularity of the value function representing Hamilton–Jacobi equation: − Ut + H(t, x, U, − Ux) = 0 with a final condition: U(T,x) = g(x). Hamilton–Jacobi equation, in which the Hamiltonian H depends on the value of solution U, is represented by the value function with more complicated structure than the value function in Bolza problem. This function is described with the use of some class of Mayer problems related to the optimal control theory and the calculus of variation. In the paper we prove that absolutely continuous functions that are solutions of Mayer problem satisfy the Lipschitz condition. Using this fact we show that the value function is a bilateral solution of Hamilton–Jacobi equation. Moreover, we prove that continuity or the local Lipschitz condition of the function of final cost g is inherited by the value function. Our results allow to state the theorem about existence and uniqueness of bilateral solutions in the class of functions that are bounded from below and satisfy the local Lipschitz condition. In proving the main results we use recently derived necessary optimality conditions of Loewen–Rockafellar [P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 32 (1994) 442–470; P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 35 (1997) 2050–2069].

A two-person zero-sum differential game with unbounded controls is considered. Underproper coercivity conditions, the upper and lower value functions are characterized as theunique viscosity solutions to the corresponding upper and lower Hamilton–Jacobi–Isaacsequations, respectively. Consequently, when the Isaacs’ condition is satisfied, the upperand lower value functions coincide, leading to the existence of the value function of thedifferential game. Due to the unboundedness of the controls, the corresponding upper andlower Hamiltonians grow super linearly in the gradient of the upper and lower valuefunctions, respectively. A uniqueness theorem of viscosity solution to Hamilton–Jacobiequations involving such kind of Hamiltonian is proved, without relying on theconvexity/concavity of the Hamiltonian. Also, it is shown that the assumed coercivityconditions guaranteeing the finiteness of the upper and lower value functions are sharp insome sense.