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Linear Equations: Square-Integrable Solutions

  • Sergey V. Lototsky
  • Boris L. Rozovsky
Chapter
Part of the Universitext book series (UTX)

Abstract

There are many standard references on SODEs and even more standard references on deterministic PDEs. Here are a few of each, listed in a non-decreasing order of difficulty:

References

  1. 5.
    A. Bain, D. Crisan, Fundamentals of Stochastic Filtering. Stochastic Modelling and Applied Probability, vol. 60 (Springer, New York, 2009)Google Scholar
  2. 11.
    D. Blömker, Nonhomogeneous noise and Q-Wiener processes on bounded domains. Stoch. Anal. Appl. 23(2), 255–273 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 14.
    B. Boufoussi, A support theorem for hyperbolic SPDEs in anisotropic Besov-Orlicz space. Random Oper. Stoch. Equ. 10(1), 59–88 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 15.
    B. Boufoussi, M. Eddahbi, M. N’zi, Freidlin-Wentzell type estimates for solutions of hyperbolic SPDEs in Besov-Orlicz spaces and applications. Stoch. Anal. Appl. 18(5), 697–722 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 16.
    R. Buckdahn, É. Pardoux, Monotonicity Methods for White Noise Driven Quasi-Linear SPDEs. Diffusion Processes and Related Problems in Analysis, Vol. I (Evanston, IL, 1989). Progress in Probability, vol. 22 (Birkhäuser Boston, Boston, 1990), pp. 219–233Google Scholar
  6. 22.
    R.A. Carmona, M.R. Tehranchi, Interest Rate Models: An Infinite Dimensional Stochastic Analysis Perspective. Springer Finance (Springer, Berlin, 2006)zbMATHGoogle Scholar
  7. 24.
    P.-L. Chow, Stochastic Partial Differential Equations. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series (Chapman & Hall/CRC, Boca Raton, 2007)zbMATHGoogle Scholar
  8. 25.
    P.-L. Chow, Y. Huang, Semilinear stochastic hyperbolic systems in one dimension. Stoch. Anal. Appl. 22(1), 43–65 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 31.
    G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, vol. 44 (Cambridge University Press, Cambridge, 1992)Google Scholar
  10. 33.
    R. Dalang, E. Nualart, Potential theory for hyperbolic SPDEs. Ann. Probab. 32(3A), 2099–2148 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 39.
    C. Donati-Martin, Quasi-linear elliptic stochastic partial differential equation: Markov property. Stoch. Stoch. Rep. 41(4), 219–240 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 40.
    C. Donati-Martin, D. Nualart, Markov property for elliptic stochastic partial differential equations. Stoch. Stoch. Rep. 46(1–2), 107–115 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 41.
    T.E. Duncan, B. Maslowski, B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise. Stoch. Process. Appl. 115(8), 1357–1383 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 42.
    N. Dunford, J.T. Schwartz, Linear Operators, I: General Theory (Wiley-Interscience, Chichester, 1988)zbMATHGoogle Scholar
  15. 45.
    M. Eddahbi, Large deviations for solutions of hyperbolic SPDE’s in the Hölder norm. Potential Anal. 7(2), 517–537 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 50.
    L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998)Google Scholar
  17. 56.
    M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 260 (Springer, New York, 1998)Google Scholar
  18. 57.
    A. Friedman, Stochastic Differential Equations and Applications (Dover Publications, Mineola, 2006)zbMATHGoogle Scholar
  19. 58.
    A. Friedman, Partial Differential Equations (Dover Publications, Mineola, 2008)Google Scholar
  20. 66.
    D. Grieser, Uniform bounds for eigenfunctions of the Laplacian on manifolds with boundary. Commun. Partial Differ. Equ. 27(7–8), 1283–1299 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 69.
    I. Gyöngy, T. Martínez, On numerical solution of stochastic partial differential equations of elliptic type. Stochastics 78(4), 213–231 (2006)MathSciNetzbMATHGoogle Scholar
  22. 77.
    E. Hille, R.S. Phillips, Functional Analysis and Semi-groups (American Mathematical Society, Providence, 1974)zbMATHGoogle Scholar
  23. 92.
    K. Iwata, The inverse of a local operator preserves the Markov property. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19(2), 223–253 (1992)Google Scholar
  24. 97.
    F. John, Partial Differential Equations, 4th edn. Applied Mathematical Sciences, vol. 1 (Springer, New York, 1991)Google Scholar
  25. 100.
    G. Kallianpur, Stochastic Filtering Theory. Applications of Mathematics, vol. 13 (Springer, Berlin, 1980)Google Scholar
  26. 103.
    I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics, vol. 113 (Springer, New York, 1991)Google Scholar
  27. 108.
    D. Khoshnevisan, Multiparameter Processes: An Introduction to Random Fields. Springer Monographs in Mathematics (Springer, New York, 2002)Google Scholar
  28. 110.
    D. Khoshnevisan, E. Nualart, Level sets of the stochastic wave equation driven by a symmetric Lévy noise. Bernoulli 14(4), 899–925 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 111.
    J.U. Kim, On a stochastic hyperbolic integro-differential equation. J. Differ. Equ. 201(2), 201–233 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 112.
    T.W. Körner, Fourier Analysis, 2nd edn. (Cambridge University Press, Cambridge, 1989)zbMATHGoogle Scholar
  31. 118.
    N.V. Krylov, Introduction to the Theory of Diffusion Processes. Translations of Mathematical Monographs, vol. 142 (American Mathematical Society, Providence, 1995)Google Scholar
  32. 119.
    N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, vol. 12 (American Mathematical Society, Providence, 1996)Google Scholar
  33. 120.
    N.V. Krylov, An Analytic Approach to SPDEs, Stochastic Partial Differential Equations, ed. by B.L. Rozovskii, R. Carmona. Six Perspectives, Mathematical Surveys and Monographs (American Mathematical Society, Providence, 1999), pp. 185–242Google Scholar
  34. 121.
    N.V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. Graduate Studies in Mathematics, vol. 96 (American Mathematical Society, Providence, 2008)Google Scholar
  35. 122.
    N.V. Krylov, B.L. Rozovskii, Stochastic evolution equations. J. Sov. Math. 14(4), 1233–1277 (1981). Reprinted in Stochastic Differential Equations: Theory and Applications, ed. by S.V. Lototsky, P.H. Baxendale. Interdisciplinary Mathematical Sciences, vol. 2 (World Scientific, 2007), pp. 1–70Google Scholar
  36. 125.
    H. Kunita, Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics, vol. 24 (Cambridge University Press, Cambridge, 1997)Google Scholar
  37. 133.
    J.-L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I (Springer, New York, 1972)Google Scholar
  38. 139.
    R.Sh. Liptser, A.N. Shiryaev, Statistics of Random Processes, I: General Theory, 2nd edn. Applications of Mathematics, vol. 5 (Springer, New York, 2001)Google Scholar
  39. 154.
    M. Marcus, V.J. Mizel, Stochastic hyperbolic systems and the wave equation. Stoch. Stoch. Rep. 36(3–4), 225–244 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 155.
    T. Martínez, M. Sanz-Solé, A lattice scheme for stochastic partial differential equations of elliptic type in dimension d ≥ 4. Appl. Math. Optim. 54(3), 343–368 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 156.
    B. Maslowski, D. Nualart, Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202(1), 277–305 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 161.
    R. Mikulevicius, On the Cauchy problem for parabolic SPDEs in Hölder classes. Ann. Probab. 28(1), 74–103 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 167.
    A. Millet, M. Sanz-Solé, The support of the solution to a hyperbolic SPDE. Probab. Theory Relat. Fields 98(3), 361–387 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 168.
    A. Millet, M. Sanz-Solé, Points of positive density for the solution to a hyperbolic SPDE. Potential Anal. 7(3), 623–659 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 172.
    E. Nelson, The free Markoff field. J. Funct. Anal. 12(2), 211–227 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 175.
    D. Nualart, The Malliavin Calculus and Related Topics, 2nd edn. (Springer, New York, 2006)zbMATHGoogle Scholar
  47. 178.
    D. Nualart, S. Tindel, Quasilinear stochastic elliptic equations with reflection. Stoch. Process. Appl. 57(1), 73–82 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 179.
    E. Nualart, F. Viens, The fractional stochastic heat equation on the circle: time regularity and potential theory. Stoch. Process. Appl. 119(5), 1505–1540 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 181.
    B.K. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 5th edn. (Springer, New York, 1998)CrossRefzbMATHGoogle Scholar
  50. 183.
    M. Ondreját, Existence of global martingale solutions to stochastic hyperbolic equations driven by a spatially homogeneous Wiener process. Stoch. Dyn. 6(1), 23–52 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 185.
    S. Peszat, J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise. Encyclopedia of Mathematics and Its Applications, vol. 113 (Cambridge University Press, Cambridge, 2007)Google Scholar
  52. 193.
    J. Rauch, Partial Differential Equations. Graduate Texts in Mathematics, vol. 128 (Springer, New York, 1991)Google Scholar
  53. 196.
    C. Rovira, M. Sanz-Solé, A nonlinear hyperbolic SPDE: approximations and support, in Stochastic Partial Differential Equations (London Mathematical Society, Edinburgh, 1994). Lecture Note Series, vol. 216 (Cambridge University Press, Cambridge, 1995), pp. 241–261Google Scholar
  54. 197.
    C. Rovira, M. Sanz-Solé, The law of the solution to a nonlinear hyperbolic SPDE. J. Theor. Probab. 9(4), 863–901 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 198.
    Yu.A. Rozanov, Random Fields and Stochastic Partial Differential Equations. Mathematics and Its Applications, vol. 438 (Kluwer Academic, Dordrecht, 1998)Google Scholar
  56. 199.
    B.L. Rozovskii, Stochastic Evolution Systems. Mathematics and Its Applications (Soviet Series), vol. 35 (Kluwer Academic, Dordrecht, 1990)Google Scholar
  57. 201.
    Yu. Safarov, D. Vassiliev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators. Translations of Mathematical Monographs, vol. 155 (American Mathematical Society, Providence, 1997)Google Scholar
  58. 205.
    M. Sanz-Solé, I. Torrecilla, A fractional Poisson equation: existence, regularity and approximations of the solution. Stoch. Dyn. 9(4), 519–548 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 206.
    M. Sanz-Solé, I. Torrecilla-Tarantino, Probability density for a hyperbolic SPDE with time dependent coefficients. ESAIM Probab. Stat. 11, 365–380 (2007) (electronic)Google Scholar
  60. 212.
    W.A. Strauss, Partial Differential Equations: An Introduction, 2nd edn. (Wiley, Chichester, 2008)zbMATHGoogle Scholar
  61. 213.
    D.W. Stroock, S.R.S. Varadhan, On the support of diffusion processes with applications to the strong maximum principle, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley, CA, 1970/1971), Vol. III: Probability Theory (University of California Press, Berkeley, CA, 1972), pp. 333–359Google Scholar
  62. 214.
    D.W. Stroock, S.R.S. Varadhan, Multidimensional Diffusion Processes (Springer, Berlin, 1979)zbMATHGoogle Scholar
  63. 216.
    M. Tehranchi, A note on invariant measures for HJM models. Finance Stochast. 9(3), 389–398 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 217.
    S. Tindel, C.A. Tudor, F. Viens, Stochastic evolution equations with fractional Brownian motion. Probab. Theory Relat. Fields 127(2), 186–204 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 223.
    J.B. Walsh, An introduction to stochastic partial differential equations, in ’Ecole d’été de probabilités de Saint-Flour, XIV—1984. Lecture Notes in Mathematics, vol. 1180 (Springer, Berlin, 1986), pp. 265–439Google Scholar
  66. 230.
    J. Xiong, An Introduction to Stochastic Filtering Theory. Oxford Graduate Texts in Mathematics, vol. 18 (Oxford University Press, Oxford, 2008)Google Scholar
  67. 231.
    K. Yosida, Functional Analysis, 6th edn. (Springer, Berlin, 1980)zbMATHGoogle Scholar
  68. 233.
    E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/A: Linear Monotone Operators (Springer, New York, 1990)CrossRefzbMATHGoogle Scholar

Copyright information

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Authors and Affiliations

  • Sergey V. Lototsky
    • 1
  • Boris L. Rozovsky
    • 2
  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA

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