# American Institute of Mathematical Sciences

• Previous Article
Global well-posedness and exponential stability for the fermion equation in weighted Sobolev spaces
• DCDS-B Home
• This Issue
• Next Article
A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents
doi: 10.3934/dcdsb.2021043
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Behavior of solution of stochastic difference equation with continuous time under additive fading noise

 Department of Mathematics, Ariel University, Ariel 40700, Israel

Received  July 2020 Revised  November 2020 Early access February 2021

Effect of additive fading noise on a behavior of the solution of a stochastic difference equation with continuous time is investigated. It is shown that if the zero solution of the initial stochastic difference equation is asymptotically mean square quasistable and the level of additive stochastic perturbations is given by square summable sequence, then the solution of a perturbed difference equation remains to be an asymptotically mean square quasitrivial. The obtained results are formulated in terms of Lyapunov functionals and linear matrix inequalities (LMIs). It is noted that the study of the situation, when an additive stochastic noise fades on the infinity not so quickly, remains an open problem.

Citation: Leonid Shaikhet. Behavior of solution of stochastic difference equation with continuous time under additive fading noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021043
##### References:

show all references

##### References:
50 trajectories (green) of the solution of the equation (1.1) by $k = 1$, $\tau = 1$, $a_0 = a_1 = b_0 = 0.45$, $b_1 = -0.45$ and $\sigma(t)$ (red)
besides of $b_1 = 0.45$">Figure 2.  The same as Fig. 1 besides of $b_1 = 0.45$
besides of $\sigma(t) = \dfrac{m}{\sqrt{t+\tau-m}}$, $t>m$">Figure 3.  The same as Fig. 1 besides of $\sigma(t) = \dfrac{m}{\sqrt{t+\tau-m}}$, $t>m$
 [1] Theresa Lange, Wilhelm Stannat. Mean field limit of Ensemble Square Root filters - discrete and continuous time. Foundations of Data Science, 2021, 3 (3) : 563-588. doi: 10.3934/fods.2021003 [2] Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074 [3] Henri Schurz. Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 353-363. doi: 10.3934/dcdss.2008.1.353 [4] Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103 [5] Julia Calatayud, Juan Carlos Cortés, Marc Jornet. On the random wave equation within the mean square context. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021082 [6] Pham Huu Anh Ngoc. New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021040 [7] Ziheng Chen, Siqing Gan, Xiaojie Wang. Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4513-4545. doi: 10.3934/dcdsb.2019154 [8] Chun-Gil Park. Stability of a linear functional equation in Banach modules. Conference Publications, 2003, 2003 (Special) : 694-700. doi: 10.3934/proc.2003.2003.694 [9] Henri Schurz. Stochastic heat equations with cubic nonlinearity and additive space-time noise in 2D. Conference Publications, 2013, 2013 (special) : 673-684. doi: 10.3934/proc.2013.2013.673 [10] Addolorata Salvatore. Sign--changing solutions for an asymptotically linear Schrödinger equation. Conference Publications, 2009, 2009 (Special) : 669-677. doi: 10.3934/proc.2009.2009.669 [11] Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521 [12] Georgios T. Kossioris, Georgios E. Zouraris. Finite element approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1845-1872. doi: 10.3934/dcdsb.2013.18.1845 [13] Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176 [14] Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 841-857. doi: 10.3934/dcdsb.2019192 [15] Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 875-887. doi: 10.3934/dcdsb.2015.20.875 [16] Ellen Baake, Michael Baake, Majid Salamat. The general recombination equation in continuous time and its solution. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 63-95. doi: 10.3934/dcds.2016.36.63 [17] Peng Cui, Hongguo Zhao, Jun-e Feng. State estimation for discrete linear systems with observation time-delayed noise. Journal of Industrial & Management Optimization, 2011, 7 (1) : 79-85. doi: 10.3934/jimo.2011.7.79 [18] Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, 2021, 11 (1) : 23-46. doi: 10.3934/mcrf.2020025 [19] Fritz Colonius, Guilherme Mazanti. Decay rates for stabilization of linear continuous-time systems with random switching. Mathematical Control & Related Fields, 2019, 9 (1) : 39-58. doi: 10.3934/mcrf.2019002 [20] Boling Guo, Guoli Zhou. On the backward uniqueness of the stochastic primitive equations with additive noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3157-3174. doi: 10.3934/dcdsb.2018305

2020 Impact Factor: 1.327