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We prove asymptotic stability of the Poisson homogeneous equilibrium amongsolutions of the Vlassov-Poisson system in the Euclidean space $\mathbb{R}^3$.More precisely, we show that small, smooth, and localized perturbations of thePoisson equilibrium lead to global solutions of the Vlasov-Poisson system,which scatter to linear solutions at a polynomial rate as $t\to\infty$.  The Euclidean problem we consider here differs significantly from theclassical work on Landau damping in the periodic setting, in several ways. Mostimportantly, the linearized problem cannot satisfy a "Penrose condition". As aresult, our system contains resonances (small divisors) and the electric fieldis a superposition of an electrostatic component and a larger oscillatorycomponent, both with polynomially decaying rates.

Nonlinear Landau damping for the Vlasov-Poisson system in $\R^3$: the Poisson equilibrium