Steady States of Fokker–Planck Equations: II. Non-existence

This is the second paper in a series concerning the study of steady states, including stationary solutions and measures, of a Fokker–Planck equation in a general domain in $$\mathbb {R}^n$$Rn with $$L^{p}_{loc}$$Llocp drift term and $$W^{1,p}_{loc}$$Wloc1,p diffusion term for any $$p>n$$p>n. In this paper, we obtain some non-existence results of stationary measures under conditions involving anti-Lyapunov type of functions associated with the stationary Fokker–Planck equation. When combined with the existence results showed in part I of the series (Huang et al. in J. Dyn Differ Equ 10.1007/s10884-015-9454-x, 2015) contained in the same volume, not only will these results yield necessary and sufficient conditions for the existence of stationary measures, but also they provide a useful tool for one to study noise perturbations of systems of ordinary differential equations, especially with respect to problems of stochastic bifurcations, as demonstrated in some examples contained in this paper. Our analysis is based on the level set method, in particular the integral identity, and measure estimates contained in our work (Huang et al. in Ann Probab 43:1712–1730, 2015).

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