The present paper is concerned with the distributional stability of a class of stochastic delay differential equations that are formulated within Hilbert spaces, which are driven by Wiener processes as well as Poisson jump processes. Utilizing the weak convergence method, sufficient conditions for this stability property are established. To illustrate the theoretical findings, two pertinent examples are presented.
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Stability in Distribution of Stochastic Delay Differential Equations Driven by Wiener Processes versus Poisson Jumps Processes in Hilbert Spaces
- Jingwen Zhang 1,*,
- Wei Wang 1,2
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Received: 13 Feb 2026 | Revised: 29 Mar 2026 | Accepted: 20 Apr 2026 | Published: 06 May 2026
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References
- 1.
Badawi, H.; Abu Arqub, O.; Shawagfeh, N. Existence, uniqueness, and numerical analysis of fractional M-derivative stochastic integrodifferential models within the shifted Legendre Gauss pseudospectral algorithm. Fluct. Noise Lett. 2025, 24, 2550055.
- 2.
Liu, K. Necessary and sufficient conditions for exponential stability and ultimate boundedness of systems governed by stochastic partial differential equations. J. London Math. Soc. 2000, 62, 311–320.
- 3.
Liu, K. Uniform stability of autonomous linear stochastic functional differential equations in infinite dimensions Stoch. Process. Appl. 2005, 115, 1131–1165.
- 4.
Sene, N. Global asymptotic stability of the fractional differential equations. J. Nonlinear Sci. Appl. 2020, 13, 171–175.
- 5.
Xie, B. The moment and almost surely exponential stability of stochastic heat equations. Proc. Am. Math. Soc. 2008, 136, 3627–3634.
- 6.
Basak, G.K.; Bhattacharya, R.N. Stability in distribution for a class of singular diffusions. Ann. Probab. 1992, 20, 312–321.
- 7.
Bao, J.H.; Yin, G.; Yuan, C.G. Stationary distributions for retarded stochastic differential equations without dissipativity. Stochastics 2017, 89, 530–549.
- 8.
Kinnally, M.S.; Williams, R.J. On existence and uniqueness of stationary distributions for stochastic delay differential equations with positivity constraints. Electron. J. Probab. 2010, 15, 409–451.
- 9.
Hu, J.; Zhang, Q.M.; Baese, A.M.; et al. Stability in distribution for a stochastic Alzheimer’s disease model with reaction diffusion. Nonlinear Dyn. 2022, 108, 4243–4260.
- 10.
Wang, Y.; Wu, F.K.; Mao, X.R. Stability in distribution of stochastic functional differential equations. Syst. Control Lett. 2019, 132, 104513.
- 11.
Fei, C.; Fei, W.Y.; Deng, S.N.; et al. Asymptotic stability in distribution of highly nonlinear stochastic differential equations with G-Brownian motion. Qual. Theory Dyn. Syst. 2023, 22, 57.
- 12.
Hu, G.X.; Wang, K. Stability in distribution of neutral stochastic functional differential equations with Markovian switching. J. Math. Anal. Appl. 2012, 385, 757–769.
- 13.
Bao, J.H.; Hou, Z.T.; Yuan, C.G. Stability in distribution of mild solutions to stochastic partial differential equations. Proc. Am. Math. Soc. 2010, 138, 2169–2180.
- 14.
Bao, J.H.; Truman, A.; Yuan, C.G. Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps. Proc. Am. Math. Soc. 2009, 465, 2111–2134.
- 15.
Gao, X.; Li, Y.; Fan, H. Stability in Distribution of Highly Nonlinear Hybrid Stochastic Delay Systems by Delay Feedback Control. IEEE Trans. Autom. Control 2024, 69, 1834–1846.
- 16.
Liu, K. Stationary distributions of second order stochastic evolution equations with memory in Hilbert spaces. Stoch. Process. Appl. 2020, 130, 366–393.
- 17.
Liu, K. Stability in distribution for stochastic differential equations with memory driven by positive semigroups and Levy processes. Appl. Math. Comput. 2019, 362, 124580.
- 18.
Tan, L.; Jin, W.; Suo, Y. Stability in distribution of neutral stochastic functional differential equations. Stat. Probab. Lett. 2015, 107, 27–36.
- 19.
Shen, G.; Wu, X.; Yin, X. Stabilization of stochastic differential equations driven by G-L´evy process with discrete-time feedback control. Discrete Contin. Dyn. Syst. B. 2021, 26, 755–774.
- 20.
Taniguchi, T. The existence and asymptotic behaviour of solutions to non-Lipschitz stochastic functional evolution equations driven by Poisson jumps. Stochastics 2010, 82, 339–363.
- 21.
Shukla, A.; Sukavanam, N.; Pandey, D.N. Controllability of Semilinear Stochastic System with Multiple Delays in Control. IFAC Proc. Vol. 2014, 47, 306–312.
- 22.
Mohammed, S.E.A. Stochastic Functional Differential Equations; Pitman Advanced Publishing Program: New York, NY, USA, 1984.
- 23.
Ikeda, N.; Watanabe, S. Stochastic Differential Equations and Diffusion Processes; North-Holland: London, UK, 1989.
- 24.
Liu, K. Stochastic Stability of Differential Equations in Abstract Spaces; Cambridge University Press: London, UK, 2019.
How to Cite
Zhang, J.; Wang, W. Stability in Distribution of Stochastic Delay Differential Equations Driven by Wiener Processes versus Poisson Jumps Processes in Hilbert Spaces. Complex Systems Stability & Control 2026, 2 (2), 7. https://doi.org/10.53941/cssc.2026.100010.
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