# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021133
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## Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise

 Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

* Corresponding author: Xiang Lv

Received  October 2020 Revised  March 2021 Early access April 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (NSFC) under Grants No.11501369, No.11771295 and No.11971316; the NSF of Shanghai under Grants No.19ZR1437100 and No. 20JC1413800; Chen Guang Project(14CG43) of Shanghai Municipal Education Commission, Shanghai Education Development Foundation; Yangfan Program of Shanghai (14YF1409100) and Shanghai Gaofeng Project for University Academic Program Development

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

Citation: Xiang Lv. Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021133
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