The Boltzmann equation is known to exhibit exponential convergence to equilibrium for small perturbations of a Maxwellian, driven by the spectral gap and coercivity properties of the linearized collision operator. In this talk, I present recent work extending this perturbative PDE theory to the Boltzmann hierarchy, an infinite system of transport–collision equations coupling all particle marginals.

By introducing a cumulant-based linearization around tensorized equilibria, I formulate a hierarchy-level hypocoercive framework that lifts coercivity, compactness, and bilinear estimates from the one-particle setting. The analysis relies on hierarchy-adapted Sobolev norms, diagonal trace estimates, and a macroscopic–microscopic decomposition. This yields global existence, uniqueness, and exponential convergence to equilibrium for small perturbations of the hierarchy.