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Approximating the coefficients in semilinear stochastic partial differential equations. (English) Zbl 1231.35324
Summary: We investigate, in the setting of UMD Banach spaces \(E\), the continuous dependence on the data \(A, F, G\) and \(\xi \) of mild solutions of semilinear stochastic evolution equations with multiplicative noise of the form \[ \begin{cases}{\mathrm d}X(t) = [AX(t) + F(t, X(t))] \, {\mathrm d}t + G(t, X(t)) \, {\mathrm d}W_H(t),\quad t \in [0,T],\\ X(0) = \xi, \end{cases} \] where \(W _{H }\) is a cylindrical Brownian motion in a Hilbert space \(H\). We prove continuous dependence of the compensated solutions \(X(t) - e ^{tA } \xi \) in the norms \(L ^{p }(\Omega ; \, C ^{\lambda }([0, T]; \, E))\) assuming that the approximating operators \(A _{n }\) are uniformly sectorial and converge to \(A\) in the strong resolvent sense and that the approximating nonlinearities \(F _{n }\) and \(G _{n }\) are uniformly Lipschitz continuous in suitable norms and converge to \(F\) and \(G\) pointwise. Our results are applied to a class of semilinear parabolic SPDEs with finite dimensional multiplicative noise.

MSC:
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35K58 Semilinear parabolic equations
60J65 Brownian motion
46B09 Probabilistic methods in Banach space theory
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References:
[1] Arendt W.: Approximation of degenerate semigroups. Taiwanese J. Math 5, 279–295 (2001) · Zbl 1025.47023
[2] W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, vol. 96 of Monographs in Mathematics, Birkhäuser Verlag, Basel, 2001. · Zbl 0978.34001
[3] Bally V, Millet A, Sanz-Solé M.: Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations. Ann. Probab 23, 178–222 (1995) · Zbl 0835.60053 · doi:10.1214/aop/1176988383
[4] J. Bierkens, Long term dynamics of stochastic evolution equations, PhD thesis, University of Leiden, 2010.
[5] Brzeźniak Z.: On stochastic convolution in Banach spaces and applications. Stochastics Stochastics Rep 61, 245–295 (1997) · Zbl 0891.60056 · doi:10.1080/17442509708834122
[6] Brzeźniak Z, Elworthy K.D: Stochastic differential equations on Banach manifolds. Methods Funct. Anal. Topology 6, 43–84 (2000) · Zbl 0965.58028
[7] Clément P, de Pagter B, Sukochev F.A, Witvliet H: Schauder decompositions and multiplier theorems. Studia Math 138, 135–163 (2000) · Zbl 0955.46004
[8] Da Prato G, Kwapień S, Zabczyk J.: Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics 23, 1–23 (1987) · Zbl 0634.60053 · doi:10.1080/17442508708833480
[9] Da Prato G, Zabczyk J: Stochastic equations in infinite dimensions, vol. 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1992) · Zbl 0762.60052
[10] Daners D.: Heat kernel estimates for operators with boundary conditions. Math. Nachr 217, 13–41 (2000) · Zbl 0973.35087 · doi:10.1002/1522-2616(200009)217:1<13::AID-MANA13>3.0.CO;2-6
[11] D. Daners, Domain perturbation for linear and semi-linear boundary value problems, in Handbook of Differential Equations, vol. 6, North-Holland, 2008, pp. 1–81. · Zbl 1197.35090
[12] Dettweiler J, Weis L.W, Neerven J.M.A.M.v: Space-time regularity of solutions of the parabolic stochastic Cauchy problem. Stoch. Anal. Appl 24, 843–869 (2006) · Zbl 1109.35124 · doi:10.1080/07362990600753577
[13] M. H. A. Haase, The functional calculus for sectorial operators, vol. 169 of Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel, 2006. · Zbl 1101.47010
[14] Hoffmann-Jørgensen J: Sums of independent Banach space valued random variables. Studia Math 52, 159–186 (1974) · Zbl 0265.60005
[15] W. B. Johnson and J. Lindenstrauss, eds., Handbook of the geometry of Banach spaces. Vol. I, North-Holland Publishing Co., Amsterdam, 2001. · Zbl 0970.46001
[16] W. B. Johnson and J. Lindenstrauss, eds., Handbook of the geometry of Banach spaces. Vol. 2, North-Holland Publishing Co., Amsterdam, 2003. · Zbl 1013.46001
[17] O. Kallenberg, Foundations of modern probability, Probability and its Applications, Springer-Verlag, New York, second ed., 2002. · Zbl 0996.60001
[18] N. J. Kalton and L. W. Weis, The H calculus and square function estimates. Preprint, 2004. · Zbl 1385.47008
[19] N. Krylov, An analytic approach to SPDEs, in in: ”Stochastic Partial Differential Equations: Six Perspectives, vol. 64 of Math. Surveys Monogr., Amer. Math. Soc., Providence, RI, 1999, pp. 185–242. · Zbl 0933.60073
[20] P. C. Kunstmann and L. W. Weis, Maximal L p -regularity for parabolic equations, Fourier multiplier theorems and H functional calculus, in Functional analytic methods for evolution equations, vol. 1855 of Lecture Notes in Math., Springer, Berlin, 2004, pp. 65–311. · Zbl 1097.47041
[21] M. C. Kunze and J. M. A. M. v. Neerven. Work in progress, 2011. · Zbl 1231.35324
[22] S. Kwapień, On Banach spaces containing c 0. Studia Math. 52 (1974), 187–188. A supplement to the paper by J. Hoffmann-Jørgensen: ”Sums of independent Banach space valued random variables” (Studia Math. 52 (1974), 159–186). · Zbl 0295.60003
[23] Marinelli C, Prévôt C, Röckner M: Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise. J. Funct. Anal 258, 616–649 (2010) · Zbl 1186.60060 · doi:10.1016/j.jfa.2009.04.015
[24] A. Millet and M. Sanz-Solé, A simple proof of the support theorem for diffusion processes, in Séminaire de Probabilités, XXVIII, vol. 1583 of Lecture Notes in Math., Springer, Berlin, 1994, pp. 36–48. · Zbl 0807.60073
[25] Neerven J.M.A.M.v: \(\gamma\)-Radonifying operators: a survey. Proceedings of the CMA 44, 1–62 (2010) · Zbl 1236.47018
[26] Neerven J.M.A.M.v, Veraar M.C, Weis L: Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal 255, 940–993 (2008) · Zbl 1149.60039 · doi:10.1016/j.jfa.2008.03.015
[27] Neerven J.M.A.M.v, Veraar M.C, Weis L.W: Stochastic integration in UMD Banach spaces. Ann. Probab 35, 1438–1478 (2007) · Zbl 1121.60060 · doi:10.1214/009117906000001006
[28] Neerven J.M.A.M.v, Weis L.W: Stochastic integration of functions with values in a Banach space. Studia Math 166, 131–170 (2005) · Zbl 1073.60059 · doi:10.4064/sm166-2-2
[29] Peszat S, Zabczyk J: Strong Feller property and irreducibility for diffusions on Hilbert spaces. Ann. Probab 23, 157–172 (1995) · Zbl 0831.60083 · doi:10.1214/aop/1176988381
[30] Seidler J: Weak convergence of infinite-dimensional diffusions. Stochastic Anal. Appl 15, 399–417 (1997) · Zbl 0886.60005 · doi:10.1080/07362999708809484
[31] Weis L.W: Operator-valued Fourier multiplier theorems and maximal L p -regularity. Math. Ann 319, 735–758 (2001) · Zbl 0989.47025 · doi:10.1007/PL00004457
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