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This paper is devoted to the study of the stochastic properties of dynamicalsystems preserving an infinite measure. More precisely we prove central limittheorems for Birkhoff sums of observables of $\mathbb{Z}^2$-extensions ofdynamical systems (satisfying some nice spectral properties). The motivation ofour paper is the $\mathbb{Z}^2$-periodic Lorentz process for which we establisha functional central limit theorem for H\"older continuous observables (indiscrete time as well as in continuous time).

Central limit theorems for the $\mathbb{Z}^2$-periodic Lorentz gas