We establish coefficient conditions for the existence and uniqueness of invariant measures in Hilbert spaces for linear stochastic equations.
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References
G. Da Plato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge (1992).
G. Da Plato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems, Cambridge Univ, Press, Cambridge (1996).
N. Kryloff and N. Bogoliouboff, “La théorie générale de la measure dans son application à l’étude des systèms dynamiques de la mécanique non linéaire,” Ann. Math. (2), 38, No. 1, 65–113 (1937).
V. I. Bogachev and M. Rockner, “Elliptic equations for measures on infinite-dimensional spaces and applications,” Probab. Theory Related Fields, 120 (4), 445–496 (2001).
G. Mueller, “Coupling and invariant measures for the heat equation with noise,” Ann. Probab., 21(4), 2189–2199 (1993).
S. Assing and R. Manthey, “Invariant measures for stochastic heat equations with unbounded coefficients,” Stoch. Proc. Appl., 103(2), 237–256 (2003).
S. Cerrai, “Stochastic reaction diffusion system with multiplicative noise and non-Lipschitz reaction term,” Probab. Theory Related Fields, 125(2), 271–304 (2003).
O. Misiats, O. Stanzhytskyi, and N. K. Yip, “Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains,” J. Theor. Probab., 29(3), 996–1026 (2015).
O. Misiats, O. Stanzhytskyi, and N. K. Yip, “Asymptotic analysis and homogenization of invariant measures,” Stochast. Dyn., 19, No. 2, 1950015 (2019).
O. Misiats, O. Stanzhytskyi, and N. K. Yip, “Invariant measures for stochastic reaction-diffusion equation with weakly dissipative nonlinearities,” Stochastics (2019); https://doi.org/10.1080/17442508.2019.1691212.
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin (1981).
E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, Clarendon Press, Oxford (1958).
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Translated from Neliniini Kolyvannya, Vol. 22, No. 4, pp. 519–525, October–December, 2019.
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Novak, I.H., Stanzhytskyi, A.O. Invariant Measures for One Class of Linear Stochastic Systems in Hilbert Spaces. J Math Sci 254, 271–279 (2021). https://doi.org/10.1007/s10958-021-05303-8
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DOI: https://doi.org/10.1007/s10958-021-05303-8