Abstract
The limiting behavior of solutions of stochastic Ito differential equations is investigated with coefficients which could be δ-type sequences or possess a degeneracy of some other form. Application of the results obtained to a study of the limiting behavior of a solution of the Cauchy problem for parabolic differential equations in partial derivatives is considered.
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G. L. Kulinich, “Necessary and sufficient conditions for convergence of solutions of one-dimensional stochastic diffusion equations under irregular dependence of coefficients on a parameter,” Teor. Veroyatn. Primen.,27, No. 4, 795–801 (1982).
G. L. Kulinich, “Asymptotic behavior of the distribution of a solution of stochastic nonhomogeneous diffusion solution,” Teor. Veroyatn. Mat. Statistika, No. 4, 95–102 (1971).
G. L. Kulinich, “On the convergence of a solution of a one-dimensional stochastic diffusion equation to the Bessel diffusion process,” in: Analit. Metody Issledov. v Teorii Veroyatnostei [in Russian], Institute of Mathematics of the Academy of Sciences of the UkrSSR, Kiev, pp. 106–113 (1980).
G. L. Kulinich, “Limiting behavior of solutions of stochastic differential equations of the diffusion type with random coefficients,” in: Predel'n. Teoremy Dlya Sluchain. Protsessov [in Russian], Institute of Mathematics of the Academy of Sciences of the UkrSSR, Kiev, pp. 137–151 (1977).
M. Almazov and G. L. Kulinich, “Asymptotic behavior of an unstable solution of the stochastic nonhomogeneous diffusion equation,” Teor. Sluchain. Protsessov,15, pp. 3–30 (1987).
N. V. Krylov, Controlled Processes of the Diffusion Type [in Russian], Nauka, Moscow (1977).
A. V. Skorokhod, Investigations in the Theory of Random Processes [in Russian], Kiev University Press, Kiev (1961).
I. P. Natanson, Theory of Functions of a Real Variable [in Russian], Nauka, Moscow (1974).
A. Yu. Veretennikov, “Strong solutions of stochastic differential equations,” Teor. Veroyatn. Primen.,24, No. 2, 348–360 (1979).
I. I. Gikhman and A. V. Skorokhod, Introduction to the Theory of Random Processes [in Russian], Nauka, Moscow (1965) (English translation: W. B. Saunders, Philadelphia, 1969).
I. I. Gikhman, “Differential equations with random functions,” in: Zimnyaya Shkola po Teorii Veroyatn. Mat. Statistike [in Russian], Institute of Mathematics of the Academy of Sciences of the UkrSSR, Kiev, pp. 42–85 (1964).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 435–443, April, 1990.
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Almazov, M., Kulinich, G.L. Limit theorems for one-dimensional nonhomogeneous stochastic diffusion equations under irregular dependence of the coefficients on a parameter. Ukr Math J 42, 383–390 (1990). https://doi.org/10.1007/BF01071322
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DOI: https://doi.org/10.1007/BF01071322