Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise

This paper is concerned with the existence of unstable stationary solutions for nonlinear stochastic differential equations (SDEs) with additive white noise. Assume that the nonlinear term \begin{document}$ f $\end{document} is monotone (or anti-monotone) and the global Lipschitz constant of \begin{document}$ f $\end{document} is smaller than the positive real part of the principal eigenvalue of the competitive matrix \begin{document}$ A $\end{document} , the random dynamical system (RDS) generated by SDEs has an unstable \begin{document}$ \mathscr{F}_+ $\end{document} -measurable random equilibrium, which produces a stationary solution for nonlinear SDEs. Here, \begin{document}$ \mathscr{F}_+ = \sigma\{\omega\mapsto W_t(\omega):t\geq0\} $\end{document} is the future \begin{document}$ \sigma $\end{document} -algebra. In addition, we get that the \begin{document}$ \alpha $\end{document} -limit set of all pull-back trajectories starting at the initial value \begin{document}$ x(0) = x\in\mathbb{R}^n $\end{document} is a single point for all \begin{document}$ \omega\in\Omega $\end{document} , i.e., the unstable \begin{document}$ \mathscr{F}_+ $\end{document} -measurable random equilibrium. Applications to stochastic neural network models are given.