Abstract
Theorems on the existence and uniqueness of the statistical solutions and for the existence of stationary solutions for nonlinear stochastic equations are proved. Markov families of statistical solutions are constructed. The ergodic properties of monotone systems are investigated. A Girsanov type theorem on the absolute continuity of the statistical solution of a nonlinear stochastic parabolic equation with respect to the statistical solution of the monotone equation is proved.
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Literature cited
M. I. Vishik, A. I. Komech, and A. V. Fursikov, “Some mathematical problems of statistical hydromechanics,” Usp. Mat. Nauk,34, No. 5, 135–210 (1979).
T. V. Girya, “Stabilization of statistical solutions of a nonlinear parabolic equations with white noise,” Usp. Mat. Nauk,36, No. 2, 177–178 (1981).
I. I. Gikhman (Gihman) and A. V. Skorokhod (Skorohod), The Theory of Stochastic Processes. I, Springer, New York (1974).
I. I. Gikhman (Gihman) and A. V. Skorokhod (Skorohod), The Theory of Stochastic Processes. II, Springer, New York (1975).
Yu. I. Daletskii and S. V. Fomin, Measures and Differential Equations in Infinite-Dimensional Spaces [in Russian], Nauka, Moscow (1983).
I. V. Girsanov, “On the transformation of a certain class of random processes by an absolutely continuous substitution of the measure,” Teor. Veroyatn. Primen.,5, No. 3, 314–330 (1960).
E. Gagliardo, “Proprieta di alcune classi di funzioni in piu variabili,” Ricerche Mat.,7, No. 1, 102–137 (1958).
J. L. Doob, Stochastic Processes, Wiley, New York (1953).
R. S. Khas'minskii, Stability of Systems of Differential Equations under Random Perturbations of Their Parameters [in Russian], Nauka, Moscow (1969).
J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod, Gauthier-Villars, Paris (1969).
H. Gajewski, K. Groger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operator-differentialgleichungem, Akademie-Verlag, Berlin (1974).
M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics [in Russian], Nauka, Moscow (1980).
I. I. Gikhman (Gihman) and A. V. Skorokhod (Skorohod), The Theory of Stochastic Processes. III, Springers New York (1979).
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Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 12, pp. 88–117, 1987.
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Il'yasov, Y.S., Komech, A.I. Girsanov's theorem and ergodic properties of statistical solutions of nonlinear parabolic equations. J Math Sci 47, 2547–2569 (1989). https://doi.org/10.1007/BF01102998
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DOI: https://doi.org/10.1007/BF01102998