The Kramers-Fokker-Planck equation with a decaying potential in Rⁿ, 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|^-) for some > 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 (4 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.