The Kramers-Fokker-Planck equation with a decaying potential in $ R^n$, $n ≥ 4$

We use methods from microlocal analysis and quantum scattering to study spectral properties near the threshold zero of the Kramers-Fokker-Planck operator with a decaying potential in Rⁿ, n ≥ 4, and deduce the large-time behavior of solutions to the kinetic Kramers-Fokker-Planck equation. For short-range potentials, we establish an optimal time-decay estimate in weighted L²-spaces when n≥ 5 is odd. For potentials decaying like O(|x|-ρ) for some ρ > n-1, we obtain, for all dimensions n ≥ 4, a large-time expansion of the solution with the leading term given by the Maxwell-Boltzmann distribution multiplied by the factor (4π t)- n 2 corresponding to the decay for the heat equation. These results complete those obtained in [16, 22] for dimensions $n=1$ and $3$. The same questions for $n=2$ are still open.