Computer Vision

2014 Edition
| Editors: Katsushi Ikeuchi

Stochastic Partial Differential Equations

  • Annika Lang
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-31439-6_681

Synonyms

Definition

A stochastic partial differential equation (SPDE) is a partial differential equation (PDE) with an extra stochastic term, e.g., an Itô integral. Sometimes partial differential equations, where the differential operator or the initial condition is disturbed, are also called SPDEs, but the more common term for these equations is random PDEs.

Background

First results on SPDEs and infinite-dimensional stochastic differential equations (SDEs) appeared in the mid-1960s. Ample publications and results are due to the end 1970s and early 1980s. Here the work by Walsh [20] and Pardoux [16] should be mentioned. In the early 1990s, Da Prato and Zabczyk published their book with an infinite-dimensional approach to SPDEs driven by Wiener processes [6]. In the last years, a number of books on SPDEs were published, in particular an extension of [6] by Peszat and Zabczyk [17]. Other recent publications to be mentioned...

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References

  1. 1.
    Barth A, Lang A (2012) Simulation of stochastic partial differential equations using finite element methods. Stochastics 84(2–3):217–231MathSciNetzbMATHGoogle Scholar
  2. 2.
    Braess D (2007) Finite elements. Theory, fast solvers and applications in elasticity theory (Finite Elemente. Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie.) 4th rev and extended edn. Springer, BerlinCrossRefGoogle Scholar
  3. 3.
    Chow P-L (2007) Stochastic partial differential equations. Applied mathematics and nonlinear science series. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  4. 4.
    Dautray R, Lions J-L (2000) Mathematical analysis and numerical methods for science and technology. Springer, BerlinCrossRefGoogle Scholar
  5. 5.
    Dellacherie C, Meyer P-A (1978) Probabilities and potential (Transl. from the French.) North-Holland mathematics studies, vol 29. North-Holland, Amsterdam/New York/OxfordGoogle Scholar
  6. 6.
    Da Prato G, Zabczyk J (1992) Stochastic equations in infinite dimensions. Encyclopedia of mathematics and its applications, vol 44. Cambridge University Press, CambridgeGoogle Scholar
  7. 7.
    Fishman GS (1996) Monte Carlo. Concepts, algorithms, and applications. Springer, New YorkGoogle Scholar
  8. 8.
    Holden H, Øksendal B, Ubøe J, Zhang T (2010) Stochastic partial differential equations. A modeling, white noise functional approach. Universitext, 2nd edn. Springer, New YorkGoogle Scholar
  9. 9.
    Ikeda N, Watanabe S (1989) Stochastic differential equations and diffusion processes. North-Holland mathematical library, 2nd edn., vol 24. North-Holland, Amsterdam/Kodansha, TokyoGoogle Scholar
  10. 10.
    Juan O, Keriven R, Postelnicu G (2006) Stochastic motion and the level set method in computer vision: stochastic active contours. Int J Comput Vis 69(1):7–25CrossRefGoogle Scholar
  11. 11.
    Kloeden PE, Platen E (1992) Numerical solution of stochastic differential equations. Applications of mathematics, vol 23. Springer, BerlinGoogle Scholar
  12. 12.
    . Lang A (2007) Simulation of stochastic partial differential equations and stochastic active contours. PhD thesis, Universität MannheimGoogle Scholar
  13. 13.
    Lions P-L, Souganidis PE (1998) Fully nonlinear stochastic partial differential equations. C R Acad Sci Paris Sér I Math 326(9):1085–1092MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Métivier M (1982) Semimartingales: a course on stochastic processes. de Gruyter studies in mathematics, vol 2. Walter de Gruyter, Berlin/New YorkGoogle Scholar
  15. 15.
    Øksendal B (2003) Stochastic differential equations. An introduction with applications. Universitext, 6th edn. Springer, BerlinGoogle Scholar
  16. 16.
    Pardoux E (1979) Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3:127–167MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Peszat S, Zabczyk J (2007) Stochastic partial differential equations with lévy noise. An evolution equation approach. Encyclopedia of mathematics and its applications, vol 113. Cambridge University Press, CambridgeGoogle Scholar
  18. 18.
    Prévôt C, Röckner M (2007) A concise course on stochastic partial differential equations. Lecture notes in mathematics, vol 1905. Springer, BerlinGoogle Scholar
  19. 19.
    Protter PE (2004) Stochastic integration and differential equations. Applications of mathematics, 2nd edn., vol 21. Springer, BerlinGoogle Scholar
  20. 20.
    Walsh JB (1986) An introduction to stochastic partial differential equations (École d'été de probabilités de Saint-Flour XIV - 1984.) Lect Notes Math 1180:265–437CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Annika Lang
    • 1
  1. 1.Seminar für Angewandte Mathematik, ETH ZürichZürichSwitzerland