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As for the spatially homogeneous Boltzmann equation of Maxwellian moleculeswith the fractional Fokker-Planck diffusion term, we consider the Cauchyproblem for its Fourier-transformed version, which can be viewed as a kineticmodel for the stochastic time-evolution of characteristic functions associatedwith the symmetric stable Levy process and the Maxwellian collision dynamics.Under a non-cutoff assumption on the kernel, we establish a global existencetheorem with maximum growth estimate, uniqueness and stability of solutions.

On the Boltzmann equation with the symmetric stable Levy process