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

Xiang Lv

doi:10.3934/dcdsb.2021133

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2022, Volume 27, Issue 4: 2313-2323. Doi: 10.3934/dcdsb.2021133

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

Xiang Lv

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

Received: September 30, 2020

Revised: February 28, 2021

Early access: April 2021

Published: April 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 37H05, 60H10; Secondary: 34K21, 34K20.

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