A Note on the instability of equilibria for distribution dependent SDEs
Shao-Qin Zhang
TL;DR
This paper addresses the instability of stationary distributions for distribution-dependent SDEs, a phenomenon tied to phase transitions in mean-field systems. It develops a spectral criterion based on the generator of the linearized nonlinear semigroup, showing that if the spectrum of $L_{\bar{A}}^*$ intersects the positive real axis, the stationary distribution $\mu_\infty$ cannot be attracting in the metric $\|\cdot\|_{V_0,\phi_0}$. The analysis relies on the quasi-compactness of the linearized semigroup $Q_t$, a nonlinear Duhamel framework, and the construction of perturbations along eigenfunctions to demonstrate exponential growth of deviations from $\mu_\infty$, thereby establishing instability. Concrete examples, notably granular-media-type models with double-well landscapes, illustrate how spectral properties govern phase transitions and ergodicity breakdown in distribution-dependent SDEs.
Abstract
Due to the existence of multiple stationary distributions, we study the stability and instability of a stationary distribution for distribution dependent stochastic differential equations. This note is devoted to the instability of a stationary distribution, and links the instability to a spectral property of the generator of the corresponding linearized semigroup to the stochastic equation. Concrete examples, such as the granular media equation with double-wells landscapes, are given to illustrate our main result.