The flow of a Hamilton-Jacobi PDE yields a dynamical system on the space of continuous functions. When the Hamiltonian function is convex in the momentum variable, and the spatial dimension is one, we may restrict the flow to piecewise smooth functions and give a kinetic description for the solution. We regard the locations of jump discontinuities of the first derivative of solutions as the sites of particles. These particles interact via collisions and coagulations. When these particles are selected randomly according to certain Gibbs measures initially, then the law of particles remains Gibbsian at later times, and one can derive a Boltzmann/Smoluchowski type PDE for the evolution of these Gibbs measures. In higher dimensions, we assume that the Hamiltonian function is independent of position and that the initial condition is piecewise linear and convex. Such initial conditions can be identified as (Laguerre) tessellations and the Hamilton-Jacobi evolution can be described as a billiard on the set of tessellations.
