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

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> is monotone (or anti-monotone) and the global Lipschitz constant of <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> is smaller than the positive real part of the principal eigenvalue of the competitive matrix <inline-formula><tex-math id="M3">\begin{document}$ A $\end{document}</tex-math></inline-formula>, the random dynamical system (RDS) generated by SDEs has an unstable <inline-formula><tex-math id="M4">\begin{document}$ \mathscr{F}_+ $\end{document}</tex-math></inline-formula>-measurable random equilibrium, which produces a stationary solution for nonlinear SDEs. Here, <inline-formula><tex-math id="M5">\begin{document}$ \mathscr{F}_+ = \sigma\{\omega\mapsto W_t(\omega):t\geq0\} $\end{document}</tex-math></inline-formula> is the future <inline-formula><tex-math id="M6">\begin{document}$ \sigma $\end{document}</tex-math></inline-formula>-algebra. In addition, we get that the <inline-formula><tex-math id="M7">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-limit set of all pull-back trajectories starting at the initial value <inline-formula><tex-math id="M8">\begin{document}$ x(0) = x\in\mathbb{R}^n $\end{document}</tex-math></inline-formula> is a single point for all <inline-formula><tex-math id="M9">\begin{document}$ \omega\in\Omega $\end{document}</tex-math></inline-formula>, i.e., the unstable <inline-formula><tex-math id="M10">\begin{document}$ \mathscr{F}_+ $\end{document}</tex-math></inline-formula>-measurable random equilibrium. Applications to stochastic neural network models are given.</p>