On the Cauchy problem for parabolic SPDEs in Hölder classes. (English) Zbl 1044.60050

Summary: We study Cauchy’s problem for certain second-order linear parabolic stochastic differential equation (SPDE) driven by a cylindrical Brownian motion. Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Hölder classes.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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[1] DaPrato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press. · Zbl 0761.60052
[2] Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ. · Zbl 0144.34903
[3] Gilbarg, D. and Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order. Springer, New York. · Zbl 0562.35001
[4] Krylov, N. V. (1996). On Lp theory of stochastic partial differential equations. SIAM J. Math. Anal. 27 313-340. · Zbl 0846.60061
[5] Krylov, N. V. and Rozovskii, B. L. (1977). On the Cauchy problem for linear partial differential equations. Math. USSR Izvestija 11 1267-1284. · Zbl 0396.60058
[6] Ladyzhenskaja, O. A., Solonnikov, V. A. and Uraltseva, N. N. (1968). Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Providence, RI. · Zbl 0174.15403
[7] Mikulevicius, R. and Pragarauskas, H. (1992). On the Cauchy problem for certain integro- differential operators in Sobolev and Hölder spaces. Lithuanian Math. J. 32 238-264. · Zbl 0795.45007
[8] Mikulevicius, R. and Rozovskii, B. L. (1998). Linear parabolic stochastic PDEs and Wiener chaos. SIAM J. Math. Anal. 29 452-480. · Zbl 0911.60045
[9] Pardoux, E. (1975). Equations aux derivées partielles stochastiques non linéaires monotones. Étude de solutions de type It o, Th ese, Univ. Paris Sud, Orsay.
[10] Rozovskii, B. L. (1975). On stochastic partial differential equations. Mat. Sbornik 96 314-341.
[11] Rozovskii, B. L. (1990). Stochastic Evolution Systems. Kluwer, Norwell. · Zbl 0724.60070
[12] Zakai, M. (1969). On the optimal filtering of diffusion processes.Wahrsch. Verw Gebiete 11 230-243. · Zbl 0164.19201
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