For the hard sphere gas, cumulants have been introduced to encode correlations between particles. In recent years, they have played a central role in the study of kinetic limits, fluctuations, and large deviations. A Hamilton–Jacobi equation was formulated to describe the evolution of these cumulants after taking the Boltzmann–Grad limit. In this talk, I will present my previous results on the global-in-time solution and long-time convergence of this H-J equation. I will also discuss some related programs in their very early stages, particularly the attempt to develop an analytical (nonlinear PDE) approach to the study of cumulants, which are highly combinatorial and geometrical in nature.