Vlasov–Fokker–Planck equation: stochastic stability of resonances and unstable manifold expansion

We investigate the dynamics close to a homogeneous stationary state of the Vlasov equation in one dimension, in presence of a small dissipation modeled by a Fokker–Planck operator. When the stationary state is stable, we show the stochastic stability of Landau poles. When the stationary state is unstable, depending on the relative size of the dissipation and the unstable eigenvalue, we find three distinct nonlinear regimes: for a very small dissipation, the system behaves as the pure Vlasov equation; for a strong enough dissipation, the dynamics presents similarities with a standard dissipative bifurcation; in addition, we identify an intermediate regime interpolating between the two previous ones. The nonlinear analysis relies on an unstable manifold expansion, performed using Bargmann representation for the functions and operators analyzed. The resulting series are estimated with Mellin transform techniques.