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A linear Boltzmann equation is interpreted as the forward equation for theprobability density of a Markov process (K(t), i(t), Y(t)), where (K(t), i(t))is an autonomous reversible jump process, with waiting times between two jumpswith finite expectation value but infinite variance, and Y(t) is an additivefunctional of K(t). We prove that under an anomalous rescaling Y converges indistribution to a two-dimensional Brownian motion. As a consequence, theappropriately rescaled solution of the Boltzmann equation converges to adiffusion equation.

From a kinetic equation to a diffusion under an anomalous scaling