
Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise
Author(s) -
Xiang Lv
Publication year - 2022
Publication title -
discrete and continuous dynamical systems. series b
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.864
H-Index - 53
eISSN - 1553-524X
pISSN - 1531-3492
DOI - 10.3934/dcdsb.2021133
Subject(s) - mathematics , lipschitz continuity , monotone polygon , combinatorics , omega , pure mathematics , physics , geometry , quantum mechanics
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.