Stability of Rayleigh–Jeans equilibria on the whole frequency space

Characterize the stability of Rayleigh–Jeans equilibria for the spatially homogeneous isotropic four-wave kinetic equation associated with the cubic nonlinear Schrödinger equation on the whole frequency space.

Background

The paper explains that Rayleigh–Jeans equilibria have heavy high-frequency tails, are not integrable, and require exact detailed-balance cancellations because the individual collision terms diverge. These features make stability analysis on the whole frequency domain substantially more difficult than perturbation theory around Maxwellian equilibria.

The paper resolves stability for sufficiently small perturbations of nonsingular Rayleigh–Jeans equilibria with positive chemical potential in a weighted L² space. The broader stability question stated in the passage remains unresolved for the full class of Rayleigh–Jeans equilibria, including regimes not covered by that perturbative result.

References

The Rayleigh-Jeans equilibria, even though fundamental thermodynamic equilibria of wave kinetic equations and analogous to the Maxwellians in the parallel kinetic theory of particles, have not been studied sufficiently in the wave setting up to date, and basic properties on the whole frequency space, such as their stability have remained open and, in fact, doubtful.

— Global Well-Posedness near Rayleigh-Jeans Equilibria for the Cubic NLS Wave Kinetic Equation  (2608.19483 - Escobedo et al., 19 Aug 2026) in Section 1, subsection “Difficulties and strategy of the proof”