Abstract
We study the Cauchy–Dirichlet problem for a second-order quasilinear parabolic stochastic differential equation (SPDE) in a domain with a zero order noise term 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 with weights.
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References
Rozovskii, B.L.: ‘On stochastic partial differential equations’, Mat. Sbornik. 96 (1975), 314–341.
Gilbarg, D. and Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Springer, Berlin Heidelberg New York, 1983.
DaPrato, G. and Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
Krylov, N.V. and Rozovskii, B.L.: ‘On the Cauchy problem for linear partial differential equations’. Math. USSR Izvestija, 11 (1977), 1267–1284.
Kim, K.-H.: ‘On stochastic PDES with variable coefficients on C 1 domains’. Stoch. Processes. Their Appl. 112(2) (2004), 281–283.
Pardoux, E.: Equations aux derivées partielles stochastiques non linéaires monotones. Étude de solutions de type Ito: Thèse, Université de Paris Sud Sud, Orsay, 1975.
Rozovskii, B.L.: Stochastic Evolution Systems, Kluwer, Norwell, 1990.
Zakai, M.: ‘On the optimal filtering of diffusion processes’, Z. Wahrsch. 11 (1969), 230–243.
Margulis, L.G.: Investigation of integral and differential non linear filtering equations for Markov processes, PhD Thesis, Kiev, 1980.
Ladyzhenskaja, O.A., Solonnikov, V.A. and Uraltseva, A.I.: ‘Linear and Quasilinear Equations of Parabolic Type’, Translations of Mathematical Monographs, 23 (1968), AMS.
Krylov, N.V.: An analytic approach to SPDEs, in: Stochastic Partial Differential Equations: Six Perspectives, AMS Mathematical Surveys and Monographs 64, 1999, pp. 185–242.
Krylov, N.V. and Lototsky, S.V.: ‘A Sobolev space theory of SPDEs with constant coefficients on a half line’, SIAM J. Math. Anal. 30(2) (1999), 298–325.
Krylov, N.V. and Lototsky, S.V.: ‘A Sobolev space theory of SPDEs with constant coefficients on a half space’, SIAM J. Math. Anal. 31(1) (1999), 19–33.
Lieberman G.M.: Second Order Parabolic Differential Equations, World Scientific, Singapore, 1996.
Lototsky, S.V.: ‘Dirichlet problem for stochastic parabolic equations in smooth domains’, Stoch. Stoch. Rep. 68 (1999), 145–175.
Mikulevicius, R.: On the Cauchy problem for parabolic SPDEs in Hölder classes’, Ann. Probab. 28(1) (2000), 74–108.
Mikulevicius, R. and Pragarauskas, H.: ‘On the Cauchy–Dirichlet problem for parabolic SPDEs in half-space in Hölder classes’, Stochastic Processes and Applications 106 (2003), pp. 185–222.
Solonnikov V.V.: ‘A priori estimates for the second order equations of parabolic type, Trudy Mat. Inst. Steklov’ 70 (1964), 133–212 (in Russian).
Stupelis, L.: Navier–Stokes Equations in Irregular Domains, Kluwer, Dordrecht, 1995.
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Mikulevicius, R., Pragarauskas, H. On Cauchy—Dirichlet Problem for Parabolic Quasilinear SPDEs. Potential Anal 25, 37–75 (2006). https://doi.org/10.1007/s11118-005-9006-9
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DOI: https://doi.org/10.1007/s11118-005-9006-9