Exponential decay of Fisher information in the entropic-gap regime

Prove that along solutions of the spatially homogeneous Boltzmann equation with collision kernels satisfying the entropic-gap condition y+ν ≥ 2, the Fisher information obeys an exponential decay estimate of the form −dI/dt ≥ K I for some K>0, thereby establishing exponential convergence of I(t).

Background

In regimes with an entropic gap (y+ν ≥ 2), exponential convergence of entropy is known, and one expects analogous exponential decay for the Fisher information. The paper derives polynomial-decay results broadly and poses whether a direct Fisher-information inequality can be proved in this stronger form in the entropic-gap region.

Establishing −dI/dt ≥ K I would sharpen the long-time behavior of Fisher information beyond the O(t^{-α}) rates obtained elsewhere, aligning it with exponential entropy production in this regime.