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We consider the long time behavior of solutions of the d-dimensional linear Boltzmann equation that arises in the weak coupling limit for the Schrödinger equation with a time-dependent random potential. We show that the intermediate mesoscopic time limit satisfies a Fokker–Planck type equation with the wave vector performing a Brownian motion on the (d ? 1)-dimensional sphere of constant energy, as in the case of a time-independent Schrödinger equation. However, the long time limit of the solution with an isotropic initial data satisfies an equation corresponding to the energy being the square root of a Bessel process of dimension d/2. 相似文献