  We study a class of Hamilton-Jacobi partial differential equations in the
space of probability measures. In the first part of this paper, we prove
comparison principles (implying uniqueness) for this class. In the second part,
we establish the existence of a solution and give a representation using a
family of partial differential equations with control. A large part of our
analysis exploits special structures of the Hamiltonian, which might look
mysterious at first sight. However, we show that this Hamiltonian structure
arises naturally as limit of Hamiltonians of microscopical models. Indeed, in
the third part of this paper, we informally derive the Hamiltonian studied
before, in a context of fluctuation theory on the hydrodynamic scale. The
analysis is carried out for a specific model of stochastic interacting
particles in gas kinetics, namely a version of the Carleman model. We use a
two-scale averaging method on Hamiltonians defined in the space of probability
measures to derive the limiting Hamiltonian.
