Abstract
We study the stability and asymptotic stability of the zero solution of an autonomous evolution stochastic differential equation with discontinuous coefficients in a Hilbert space. The second Lyapunov method is applied to this end. An analog of the Lyapunov stability theorem, as well as an analog of the Barbashin–Krasovskii theorem on the asymptotic stability of the zero solution, is resulting for the equations in question.
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REFERENCES
Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions, Cambridge: Cambridge Univ. Press, 2014.
Liu, K., On stability for a class of semilinear stochastic evolution equations, Stochastic Process. Their Appl., 1997, vol. 70, pp. 219–241.
Ichikawa, A., Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl., 1982, vol. 90, pp. 12–44.
Vas’kovskii, M.M. and Kachan, I.V., Stability of solutions of stochastic functional-differential equations with locally Lipschitz coefficients in Hilbert spaces, Differ. Equations, 2018, vol. 54, no. 7, pp. 845–859.
Levakov, A.A., Stokhasticheskie differentsial’nye uravneniya (Stochastic Differential Equations), Minsk: Belarus. Gos. Univ., 2009.
Levakov, A.A., Stability analysis of stochastic differential equations with the use of Lyapunov functions of constant sign, Differ. Equations, 2011, vol. 47, no. 9, pp. 1271–1280.
Vas’kovskii, M.M., Zadvorny, Ya.B., and Kachan, I.V., Studying the stability of solutions of nonautonomous stochastic differential equations with discontinuous coefficients using the method of Lyapunov functions, Vestn. Belarus. Gos. Univ. Ser. 1 , 2015, no. 3, pp. 117–125.
Zadvorny, Ya.B., Global stability of an autonomous stochastic delay differential equation with discontinuous coefficients, Differ. Equations, 2018, vol. 54, no. 6, pp. 716–727.
Vas’kovskii, M.M., Existence of \(\beta \)-martingale solutions of stochastic evolution functional equations of parabolic type with measurable locally bounded coefficients, Differ. Equations, 2012, vol. 48, no. 8, pp. 1065–1080.
Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of Function Theory and Functional Analysis), Moscow: Nauka, 1976.
Levakov, A.A. and Vas’kovskii, M.M., Stokhasticheskie differentsial’nye uravneniya i vklyucheniya (Stochastic Differential Equations and Inclusions), Minsk: Belarus. Gos. Univ., 2019.
Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion Processes, Amsterdam–Tokyo: North Holland/Kodansha, 1981.
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Translated by V. Potapchouck
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Zadvorny, Y.B. Global Stability of a Stochastic Differential Equation with Discontinuous Coefficients in a Hilbert Space. Diff Equat 56, 543–557 (2020). https://doi.org/10.1134/S0012266120050018
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DOI: https://doi.org/10.1134/S0012266120050018