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

## TL;DR

This paper studies the stability and dynamics near stationary states of the Vlasov equation with dissipation, revealing regimes of stability, bifurcation, and intermediate behavior through unstable manifold analysis.

## Contribution

It introduces an unstable manifold expansion method using Bargmann representation and Mellin transforms to analyze nonlinear regimes of the Vlasov-Fokker-Planck equation.

## Key findings

- Stochastic stability of Landau poles in stable states.
- Identification of three nonlinear regimes based on dissipation strength.
- Development of a novel unstable manifold expansion technique.

## Abstract

We investigate the dynamics close to a homogeneous stationary state of 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 a 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 non linear 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.