Skip to main content
SpringerLink
Log in
Book cover

Stochastic Partial Differential Equations pp 75–142Cite as

Stochastic Analysis in Infinite Dimensions

  • Chapter
  • First Online:

  • 4352 Accesses

Part of the book series: Universitext ((UTX))

Abstract

This chapter contains somewhat abstract but necessary, material on functional analysis and stochastic calculus. To save time, one can move on to the following chapters and come back as necessary.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   64.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   84.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. N. Aronszajn, Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  2. R.M. Balan, B.G. Ivanoff, A Markov property for set-indexed processes. J. Theor. Probab. 15(3), 553–588 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. V.I. Bogachev, Gaussian Measures. Mathematical Surveys and Monographs, vol. 62 (American Mathematical Society, Providence, 1998)

    Google Scholar 

  4. P.-L. Chow, Stochastic Partial Differential Equations. Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series (Chapman & Hall/CRC, Boca Raton, 2007)

    MATH  Google Scholar 

  5. 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 

  6. R.C. Dalang, J.B. Walsh, The sharp Markov property of Lévy sheets. Ann. Probab. 20(2), 591–626 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  7. R.C. Dalang, J.B. Walsh, The sharp Markov property of the Brownian sheet and related processes. Acta Math. 168(3–4), 153–218 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. N. Dunford, J.T. Schwartz, Linear Operators, I: General Theory (Wiley-Interscience, Chichester, 1988)

    MATH  Google Scholar 

  9. N. Dunford, J.T. Schwartz, Linear Operators, II: Spectral Theory (Wiley-Interscience, Chichester, 1988)

    MATH  Google Scholar 

  10. L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, 1998)

    Google Scholar 

  11. I.M. Gelfand, N.Ya. Vilenkin, Generalized Functions, vol. 4: Applications of Harmonic Analysis (Academic, New York, 1964)

    Google Scholar 

  12. 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 

  13. O. Kallenberg, R. Sztencel, Some dimension-free features of vector-valued martingales. Probab. Theory Relat. Fields 88, 215–247 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd edn. Graduate Texts in Mathematics, vol. 113 (Springer, New York, 1991)

    Google Scholar 

  15. K. Karhunen, Zur Spektraltheorie stochastischer Prozesse. Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys. 34, 1–7 (1946)

    MathSciNet  MATH  Google Scholar 

  16. K. Karhunen, Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys. 37, 3–79 (1947)

    MathSciNet  MATH  Google Scholar 

  17. S.G. Kreĭn, Yu.Ī. Petunı̄n, E.M. Semënov, Interpolation of Linear Operators. Translations of Mathematical Monographs, vol. 54 (American Mathematical Society, Providence, 1982)

    Google Scholar 

  18. N.V. Krylov, Introduction to the Theory of Diffusion Processes. Translations of Mathematical Monographs, vol. 142 (American Mathematical Society, Providence, 1995)

    Google Scholar 

  19. 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. 1–70

    Google Scholar 

  20. H. Kunita, Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics, vol. 24 (Cambridge University Press, Cambridge, 1997)

    Google Scholar 

  21. H. Künsch, Gaussian Markov random fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26(1), 53–73 (1979)

    MathSciNet  MATH  Google Scholar 

  22. P.D. Lax, A.N. Milgram, Parabolic equations, in Contributions to the Theory of Partial Differential Equations. Annals of Mathematics Studies, vol. 33 (Princeton University Press, Princeton, 1954), pp. 167–190

    Google Scholar 

  23. P. Lévy, Sur le mouvement brownien dépendant de plusieurs paramètres. C. R. Acad. Sci. Paris 220, 420–422 (1945)

    MathSciNet  MATH  Google Scholar 

  24. R.Sh. Liptser, A.N. Shiryaev, Theory of Martingales. Mathematics and Its Applications (Soviet Series), vol. 49 (Kluwer Academic, Dordrecht, 1989)

    Google Scholar 

  25. M. Loève, Sur les fonctions aléatoires stationnaires de second ordre. Revue Sci. 83, 297–303 (1945)

    MathSciNet  MATH  Google Scholar 

  26. M. Loève, Quelques propriétés des fonctions aléatoires de second ordre. C. R. Acad. Sci. Paris 222, 469–470 (1946)

    MathSciNet  MATH  Google Scholar 

  27. H.P. McKean, Brownian motion with a several-dimensional time. Teor. Verojatnost. i Primenen. 8, 357–378 (1963)

    MathSciNet  MATH  Google Scholar 

  28. M. Métivier, J. Pellaumail, Stochastic Integration (Academic, New York, 1980)

    MATH  Google Scholar 

  29. G.J. Murphy, C ∗ -Algebras and Operator Theory (Academic, New York, 1990)

    MATH  Google Scholar 

  30. D. Nualart, Propriedad de markov para functiones aleatorias Gaussianas. Cuadern. Estadistica Mat. Univ. Granada Ser. A Prob. 5, 30–43 (1980)

    Google Scholar 

  31. D. Nualart, The Malliavin Calculus and Related Topics, 2nd edn. (Springer, New York, 2006)

    MATH  Google Scholar 

  32. L.D. Pitt, A Markov property for Gaussian processes with a multidimensional parameter. Arch. Rational Mech. Anal. 43, 367–391 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  33. C. Prévôt, M. Röckner, A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905 (Springer, Berlin, 2007)

    Google Scholar 

  34. P. Protter, Stochastic Integration and Differential Equations, 2nd edn. (Springer, New York, 2004)

    MATH  Google Scholar 

  35. B.L. Rozovskii, Stochastic Evolution Systems. Mathematics and Its Applications (Soviet Series), vol. 35 (Kluwer Academic, Dordrecht, 1990)

    Google Scholar 

  36. F. Russo, Étude de la propriété de Markov étroite en relation avec les processus planaires à accroissements indépendants, in Seminar on Probability, XVIII. Lecture Notes in Mathematics, vol. 1059 (Springer, Berlin, 1984), pp. 353–378

    Google Scholar 

  37. M.A. Shubin, Pseudo-Differential Operators and Spectral Theory, 2nd edn. (Springer, New York, 2001)

    Book  MATH  Google Scholar 

  38. 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. 265–439

    Google Scholar 

  39. J. Weidmann, Linear Operators in Hilbert Spaces. Graduate Texts in Mathematics, vol. 68 (Springer, New York, 1980)

    Google Scholar 

  40. E. Wong, Homogeneous Gauss-Markov random fields. Ann. Math. Stat. 40, 1625–1634 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  41. K. Yosida, Functional Analysis, 6th edn. (Springer, Berlin, 1980)

    MATH  Google Scholar 

  42. E. Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed-Point Theorems (Springer, New York, 1986)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, California, USA

    Sergey V. Lototsky

  2. Division of Applied Mathematics, Brown University, Providence, Rhode Island, USA

    Boris L. Rozovsky

Authors
  1. Sergey V. Lototsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Boris L. Rozovsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Lototsky, S.V., Rozovsky, B.L. (2017). Stochastic Analysis in Infinite Dimensions. In: Stochastic Partial Differential Equations. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-58647-2_3

Download citation

Publish with us

Policies and ethics

Search

Navigation