Besov regularity of stochastic partial differential equations on bounded Lipschitz domains

This thesis is concerned with the regularity of (semi-)linear second order parabolic stochastic partial differential equations (SPDEs, for short) of Itô type on bounded Lipschitz domains. The so-called adaptivity scale of Besov spaces is used to measure the regularity of the solution with respect to...

Ausführliche Beschreibung

Gespeichert in:
1. Verfasser: Cioica, Petru A.
Beteiligte: Dahlke, Stephan (Prof. Dr.) (BetreuerIn (Doktorarbeit))
Format: Dissertation
Veröffentlicht: Philipps-Universität Marburg 2014
Reine und Angewandte Mathematik
Online Zugang:PDF-Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!


2. [75] , A weighted Sobolev space theory of parabolic stochastic PDEs on non-smooth do- mains, J. Theoret. Probab. 27 (2012), no. 1, 107–136.

3. Mihály Kovács, Stig Larsson, and Karsten Urban, On wavelet-Galerkin methods for semi- linear parabolic equations with additive noise, Preprint, arXiv:1208.0433v1, 2012.

4. Jan M.A.M. van Neerven and Lutz Weis, Stochastic integration of functions with values in a Banach space, Studia Math. 166 (2005), no. 2, 131–170.

5. Albert Cohen, Numerical Analysis of Wavelet Methods, 1st ed., Studies in Mathematics and its Applications, vol. 32, Elsevier, Amsterdam, 2003.

6. Donald L. Burkholder, Martingales and singular integrals in Banach spaces, in: Handbook of the Geometry of Banach Spaces (William B. Johnson and Joram Lindenstrauss, eds.), vol. 1, Elsevier, Amsterdam, 2001–2003, pp. 233–269.

7. [84] , Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Grad. Stud. Math., vol. 96, AMS, Providence, RI, 2008.

8. Giuseppe Da Prato and Jerzy Zabczyk, Stochastic Equations in Infinite Dimensions, En- cyclopedia Math. Appl., vol. 45, Cambridge Univ. Press, Cambridge, 1998.

9. Lawrence C. Evans and Ronald F. Gariepy, Measure Theory and Fine Properties of Func- tions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1992.

10. Peter Oswald, Multilevel Finite Element Approximation. Theory and Applications, Teubner-Skripten zur Numerik, Teubner, Stuttgart, 1994.

11. Alessandra Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progr. Nonlinear Differential Equations Appl., vol. 16, Birkhäuser, Basel–Boston–Berlin, 1995.

12. Wolfgang Dahmen, Wavelet and multiscale methods for operator equations, Acta Numer. 6 (1997), 55–228.

13. Zdziss law Brze´Brze´zniak, Jan M.A.M. van Neerven, Mark C. Veraar, and Lutz Weis, Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation, J. Dif- ferential Equations 245 (2008), no. 1, 30–58.

14. Felix Abramovich, Theofanis Sapatinas, and Bernard W. Silverman, Wavelet thresholding via a Bayesian approach, J. R. Stat. Soc. Ser. B Stat. Methodol. 60 (1998), no. 4, 725–749.

15. Sergey V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations, Methods Appl. Anal. 7 (2000), no. 1, 195–204.

16. [86] , A Sobolev space theory of SPDEs with constant coefficients in a half space, SIAM J. Math. Anal. 31 (1999), no. 1, 19–33.

17. [83] , Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces, J. Funct. Anal. 183 (2001), no. 1, 1–41.

18. Claudio Canuto, Anita Tabacco, and Karsten Urban, The wavelet element method. Part I. Construction and analysis, Appl. Comput. Harmon. Anal. 6 (1999), no. 1, 1–52.

19. Petru A. Cioica, Stephan Dahlke, Stefan Kinzel, Felix Lindner, Thorsten Raasch, Klaus Ritter, and René L. Schilling, Spatial Besov regularity for stochastic partial differential equations on Lipschitz domains, Studia Math. 207 (2011), no. 3, 197–234.

20. Jesús Suárez and Lutz Weis, Addendum to " Interpolation of Banach spaces by the γ- method " , Extracta Math. 24 (2009), no. 3, 265–269.

21. David Jerison and Carlos E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), no. 1, 161–219.

22. Nicolai V. Krylov, A W n 2 -theory of the Dirichlet problem for SPDEs in general smooth domains, Probab. Theory Related Fields 98 (1994), no. 3, 389–421.

23. Hugo Aimar and Ivana Gómez, Parabolic Besov regularity for the heat equation, Constr. Approx. 36 (2012), no. 1, 145–159.

24. Petru A. Cioica and Stephan Dahlke, Spatial Besov regularity for semilinear stochastic partial differential equations on bounded Lipschitz domains, Int. J. Comput. Math. 89 (2012), no. 18, 2443–2459.

25. Franco Flandoli, Dirichlet boundary value problem for stochastic parabolic equations: com- patibility relations and regularity of solutions, Stochastics 29 (1990), no. 3, 331–357.

26. [16] , On stochastic convolution in Banach spaces and applications, Stochastics Stochas- tics Rep. 61 (1997), no. 3, 245–295.

27. Lars I. Hedberg and Yuri Netrusov, An Axiomatic Approach to Function Spaces, Spectral Synthesis, and Luzin Approximation, Mem. Amer. Math. Soc., vol. 188, AMS, Providence, RI, 2007.

28. Christoph Schwab and Rob Stevenson, Space-time adaptive wavelet methods for parabolic evolution problems, Math. Comp. 78 (2009), no. 267, 1293–1318.

29. Vyacheslav S. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains, J. Lond. Math. Soc. (2) 60 (1999), no. 1, 237–257.

30. Philippe G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathe- matics and its Applications, vol. 4, North-Holland, Amsterdam–New York–Oxford, 1978.

31. Pierre Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24, Pitman, Boston–London–Melbourne, 1985. [58] , Singularities in boundary value problems, Recherches en mathématiques ap- pliquées, vol. 22, Masson, Paris, Springer, Berlin, 1992.

32. Kyeong-Hun Kim and Nicolai V. Krylov, On the Sobolev space theory of parabolic and elliptic equations in C 1 domains, SIAM J. Math. Anal. 36 (2004), no. 2, 618–642.

33. Nicolai V. Krylov and Sergey V. Lototsky, A Sobolev space theory of SPDE with constant coefficients on a half line, SIAM J. Math. Anal. 30 (1999), no. 2, 298–325.

34. Petru A. Cioica, Kyeong-Hun Kim, Kijung Lee, and Felix Lindner, On the L q (L p )- regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains, Electron. J. Probab. 18 (2013), no. 82, 1–41.

35. Pascal Auscher, Alan McIntosh, and Andrea R. Nahmod, Holomorphic functional calculi of operators, quadratic estimates and interpolation, Indiana Univ. Math. J. 46 (1997), no. 2, 375–403.

36. Ronald A. DeVore, Björn Jawerth, and Vasil Popov, Compression of wavelet decomposi- tions, Amer. J. Math. 114 (1992), no. 4, 737–785.

37. Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Philippe Tchamitchian, The solution of the Kato square root problem for second order elliptic op- erators on R n , Ann. of Math. (2) 156 (2002), no. 2, 633–654.

38. Wolfgang Hackbusch, Elliptic Differential Equations. Theory and Numerical Treatment, Springer Ser. Comput. Math., vol. 18, Springer, Berlin–Heidelberg–New York, 1992.

39. Klaus-Jochen Engel and Rainer Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., vol. 194, Springer, New York, 2000.

40. Stephan Dahlke, Massimo Fornasier, Thorsten Raasch, Rob Stevenson, and Manuel Werner, Adaptive frame methods for elliptic operator equations: the steepest descent ap- proach, IMA J. Numer. Anal. 27 (2007), no. 4, 717–740.

41. Zdziss law Brze´Brze´zniak, Stochastic partial differential equations in M-type 2 Banach spaces, Potential Anal. 4 (1995), no. 1, 1–45.

42. Pascal Auscher and Philippe Tchamitchian, Square roots of elliptic second order divergence operators on strongly Lipschitz domains: L p theory, Math. Ann. 320 (2001), no. 3, 577–623.

43. Petru A. Cioica, Stephan Dahlke, Nicolas Döhring, Stefan Kinzel, Felix Lindner, Thorsten Raasch, Klaus Ritter, and René L. Schilling, Adaptive wavelet methods for the stochastic Poisson equation, BIT 52 (2012), no. 3, 589–614.

44. Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math., vol. 43, Cambridge Univ. Press, Cambridge, 1995.

45. Claudia Prévôt and Michael Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Math., vol. 1905, Springer, Berlin–Heidelberg, 2007.

46. Manuel Werner, Adaptive Wavelet Frame Domain Decomposition Methods for Elliptic Operator Equations, PhD thesis, Philipps-Universität Marburg, Logos, Berlin, 2009.

47. [29] , Adaptive wavelet methods II–beyond the elliptic case, Found. Comput. Math. 2 (2002), no. 3, 203–245.

48. Herbert Amann, Ordinary Differential Equations. An Introduction to Nonlinear Analysis, De Gruyter Studies in Mathematics, vol. 13, de Gruyter, Berlin–New York, 1990. [8] , Linear and Quasilinear Parabolic Problems. Volume I. Abstract Linear Theory, Monographs in Mathematics, vol. 89, Birkhäuser, Basel–Boston–Berlin, 1995.

49. [73] , An L p -theory of SPDEs on Lipschitz domains, Potential Anal. 29 (2008), no. 3, 303–326.

50. Felix Lindner, Approximation and Regularity of Stochastic PDEs, PhD thesis, TU Dresden, Shaker, Aachen, 2011.

51. Stephan Dahlke and Winfried Sickel, Besov regularity for the Poisson equation in smooth and polyhedral cones, in: Sobolev Spaces in Mathematics II, Applications to Partial Differ- ential Equations (Vladimir G. Maz'ya, ed.), International Mathematical Series 9, Springer, jointly published with Tamara Rozhkovskaya Publisher, Novosibirsk, 2008, pp. 123–145.

52. [36] , Besov regularity of edge singularities for the Poisson equation in polyhedral do- mains, Numer. Linear Algebra Appl. 9 (2002), no. 6-7, 457–466.

53. Albert Cohen, Ingrid Daubechies, and Jean-Christophe Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), no. 5, 485–560.

54. Piotr Hajj lasz, Change of variables formula under minimal assumptions, Colloq. Math. 64 (1993), no. 1, 93–101.

55. Gustavo A. Muñoz, Yannis Sarantopoulos, and Andrew Tonge, Complexifications of real Banach spaces, polynomials and multilinear maps, Studia Math. 134 (1999), no. 1, 1–33.

56. Stephan Dahlke, Wavelets: Construction Principles and Applications to the Numerical Treatment of Operator Equations, Habilitation thesis, RWTH Aachen, Berichte aus der Mathematik, Shaker, Aachen, 1997. [34] , Besov regularity for second order elliptic boundary value problems with variable coefficients, Manuscripta Math. 95 (1998), no. 1, 59–77.

57. Vasile I. Istr˘ atat¸escu, Fixed Point Theory: An Introduction, Mathematics and its Applica- tions, vol. 7, Kluwer, Dordrecht, 1981.

58. Alf Jonsson and Hans Wallin, Function Spaces on Subsets of R n , Mathematical Reports, vol. 2, pt. 1, Harwood Academic Publishers, Chur, 1984.

59. Manfred Dobrowolski, Angewandte Funktionalanalysis. Funktionalanalysis, Sobolev- Räume und Elliptische Differentialgleichungen, Springer, Berlin, 2006.

60. Nigel J. Kalton, Svitlana Mayboroda, and Marius Mitrea, Interpolation of Hardy-Sobolev- Besov-Triebel-Lizorkin spaces and applications to problems in partial differential equations, in: Interpolation Theory and Applications (Laura De Carli and Mario Milman, eds.), Contemporary Mathematics, vol. 445, AMS, Providence, RI, 2007, pp. 121–177.

61. Jöran Bergh and Jörgen Löfström, Interpolation Spaces. An Introduction, Grundlehren der mathematischen Wissenschaften, vol. 223, Springer, Berlin–Heidelberg–New York, 1976. BIBLIOGRAPHY

62. Sophie Dispa, Intrinsic characterizations of Besov spaces on Lipschitz domains, Math. Nachr. 260 (2003), no. 1, 21–33.

63. [82] , SPDEs in L q ((0, τ ], L p ) spaces, Electron. J. Probab. 5 (2000), no. 13, 1–29.

64. Ludwig Arnold, Mathematical models of chemical reactions, in: Stochastic Systems: The Mathematics of Filtering and Identification and Applications (Michiel Hazewinkel and Jan C. Willems, eds.), NATO Advanced Study Institutes Series, Series C -Mathematical and Physical Sciences, vol. 78, Springer Netherlands, Dordrecht, 1981, pp. 111–134.

65. [121] , Maximal L p -regularity for stochastic evolution equations, SIAM J. Math. Anal. 44 (2012), no. 3, 1372–1414.

66. Ronald A. DeVore, Nonlinear approximation, Acta Numer. 7 (1998), 51–150.

67. Stephan Dahlke, Wolfgang Dahmen, and Ronald A. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in: Multiscale Wavelet Meth- ods for Partial Differential Equations (Wolfgang Dahmen, Andrew J. Kurdila, and Peter Oswald, eds.), Wavelet Analysis and Its Applications, vol. 6, Academic Press, San Diego, 1997, pp. 237–283.

68. Markus Hansen, n-term approximation rates and Besov regularity for elliptic PDEs on polyhedral domains, Preprint, DFG-SPP 1324 Preprint Series, no. 131, http://www., 2012, last accessed: 17 De- cember 2013.

69. [79] , On L p -theory of stochastic partial differential equations in the whole space, SIAM J. Math. Anal. 27 (1996), no. 2, 313–340.

70. Kyeong-Hun Kim, On stochastic partial differential equations with variable coefficients in C 1 domains, Stochastic Process. Appl. 112 (2004), no. 2, 261–283.

71. Petru A. Cioica, Stephan Dahlke, Nicolas Döhring, Ulrich Friedrich, Felix Lindner, Thorsten Raasch, Klaus Ritter, and René L. Schilling, On the convergence analysis of Rothe's method, Preprint, DFG-SPP 1324 Preprint Series, no. 124, http://www., 2012, last acessed: 17 Decem- ber 2013.

72. Hans Johnen and Karl Scherer, On the equivalence of the K-functional and moduli of continuity and some applications, in: Constructive Theory of Functions of Several Vari- ables (Walter Schempp and Karl Zeller, eds.), Lecture Notes in Math., vol. 571, Springer, Heidelberg, 1977, pp. 119–140.

73. Jan Rosi´Rosi´nski and Zdiss law Suchanecki, On the space of vector-valued functions integrable with respect to the white noise, Colloq. Math. 43 (1980), 183–201.

74. Alan McIntosh, Operators which have an H ∞ functional calculus, in: Miniconference on Operator Theory and Partial Differential Equations (Canberra) (Brian Jefferies, Alan McIntosh, and Werner J. Ricker, eds.), Proceedings of the Centre for Mathematical Anal- ysis, vol. 14, ANU, 1987, pp. 210–231.

75. David Albrecht, Xuan Duong, and Alan McIntosh, Operator theory and harmonic analysis, in: Instructional Workshop on Analysis and Geometry (Canberra) (Tim Cranny and John Hutchinson, eds.), Proceedings of the Centre for Mathematics and its Applications, vol. 34, ANU, 1996, pp. 77–136.

76. Hugo Aimar, Ivana Gómez, and Bibiana Iaffei, Parabolic mean values and maximal esti- mates for gradients of temperatures, J. Funct. Anal. 255 (2008), no. 8, 1939–1956. [5] , On Besov regularity of temperatures, J. Fourier Anal. Appl. 16 (2010), no. 6, 1007–1020.

77. Lawrence C. Evans, Partial Differential Equations, Grad. Stud. Math., vol. 19, AMS, Providence, RI, 2002.

78. Gilles Pisier, Probabilistic methods in the geometry of Banach spaces, in: Probability and Analysis (Giorgio Letta and Maurizio Pratelli, eds.), Lecture Notes in Math., vol. 1206, Springer, Berlin–Heidelberg–New York, 1986, pp. 167–241.

79. [119] , γ-radonifying operators – a survey, in: The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis (Andrew Hassel, Alan McIntosh, and Robert Taggart, eds.), Proceedings of the Centre for Mathematics and its Applications, vol. 44, 2010, pp. 1–62.

80. István Gyöngy and Annie Millet, Rate of convergence of space time approximations for stochastic evolution equations, Potential Anal. 30 (2009), no. 1, 29–64.

81. Amnon Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., vol. 44, Springer, New York, 1983.

82. Elias M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser., vol. 30, Princeton Univ. Press, Princeton, NJ, 1970.

83. Thomas Runst and Winfried Sickel, Sobolev Spaces of Fractional Order, Nemytskij Op- erators, and Nonlinear Partial Differential Equations, De Gruyter Ser. Nonlinear Anal. Appl., vol. 3, de Gruyter, Berlin, 1996.

84. [74] , Sobolev space theory of SPDEs with continuous or measurable leading coefficients, Stochastic Process. Appl. 119 (2009), no. 1, 16–44.

85. Boris L. Rozovskii, Stochastic Evolution Systems. Linear Theory and Applications to Non- Linear Filtering, Mathematics and its Applications (Soviet Series), vol. 35, Kluwer, Dor- drecht, 1990.

86. [122] , Stochastic maximal L p -regularity, Ann. Probab. 40 (2012), no. 2, 788–812.

87. Jørgen Hoffmann-Jørgensen, Sums of independent Banach space valued random variables, Studia Math. 52 (1974), no. 2, 159–186.

88. Jindřich Nečas, Sur une méthode pour résoudre lesleséquations aux dérivées partielles du type elliptique, voisine de la variationnelle, Ann. Sc. Norm. Super. Pisa, Cl. Sci., III. Ser. 16 (1962), no. 4, 305–326.

89. Markus Haase, The Functional Calculus for Sectorial Operators, Oper. Theory Adv. Appl., vol. 169, Birkhäuser, Basel–Boston–Berlin, 2006.

90. Nigel J. Kalton and Lutz Weis, The H ∞ -calculus and sums of closed operators, Math. Ann. 321 (2001), no. 2, 319–345.

91. Hans Triebel, Theory of Function Spaces, Monog. Math., vol. 78, Birkhäuser, Basel– Boston–Stuttgart, 1983. BIBLIOGRAPHY [116] , Interpolation Theory, Function Spaces, Differential Operators, 2nd ed., Barth, Heidelberg–Leipzig, 1995.

92. [117] , Theory of Function Spaces III, Monogr. Math., vol. 100, Birkhäuser, Basel– Boston–Berlin, 2006.

93. Yves Meyer, Wavelets and Operators. Translated by D.H. Salinger, Cambridge Stud. Adv. Math., vol. 37, Cambridge Univ. Press, Cambridge, 1992.

94. George C. Kyriazis, Wavelet coefficients measuring smoothness in H p (R d ), Appl. Comput. Harmon. Anal. 3 (1996), no. 2, 100–119.

95. [44] , Wavelets on manifolds I: Construction and domain decomposition, SIAM J. Math.

96. Wolfgang Dahmen and Reinhold Schneider, Wavelets with complementary boundary con- ditions – function spaces on the cube, Results Math. 34 (1998), no. 3-4, 255–293.

97. Albert Cohen, Wolfgang Dahmen, and Ronald A. DeVore, Adaptive wavelet methods for elliptic operator equations: Convergence rates, Math. Comp. 70 (2001), no. 233, 27–75.

98. [35] , Besov regularity for elliptic boundary value problems in polygonal domains, Appl. Math. Lett. 12 (1999), no. 6, 31–36.

99. Osvaldo Mendez and Marius Mitrea, The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations, J. Fourier Anal. Appl. 6 (2000), no. 5, 503–531.

100. Sonja G. Cox, Stochastic Differential Equations in Banach Spaces. Decoupling, Delay Equations, and Approximations in Space and Time, PhD thesis, Technische Universiteit Delft, 2012.

101. [43] , Composite wavelet bases for operator equations, Math. Comp. 68 (1999), no. 228, 1533–1567.

102. Natalia Bochkina, Besov regularity of functions with sparse random wavelet coefficients, Preprint,, 2006, last accessed: 17 December 2013.

103. Werner Linde and Albrecht Pietsch, Mappings of Gaussian cylindrical measures in Banach spaces, Theory Probab. Appl. 19 (1974), no. 3, 445–460.

104. Michael Frazier and Björn Jawerth, A discrete transform and decompositions of distribu- tion spaces, J. Funct. Anal. 93 (1990), no. 1, 34–170.

105. [20] , The wavelet element method. Part II. Realization and additional features in 2D and 3D, Appl. Comput. Harmon. Anal. 8 (2000), no. 2, 123–165.

106. Staniss law Kwapie´Kwapie´n, On Banach spaces containing c 0 . A supplement to the paper by J. Hoffmann-Jørgensen " Sums of independent Banach space valued random variables " , Studia Math. 52 (1974), no. 2, 187–188.