Skip to main content
SpringerLink
Log in

A Primer on Stochastic Partial Differential Equations

  • Chapter
A Minicourse on Stochastic Partial Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1962))

These notes form a brief introductory tutorial to elements of Gaussian noise analysis and basic stochastic partial differential equations (SPDEs) in general, and the stochastic heat equation, in particular. The chief aim here is to get to the heart of the matter quickly. We achieve this by studying a few concrete equations only. This chapter provides sufficient preparation for learning more advanced theory from the remainder of this volume.

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

Access this chapter

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 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 44.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Louis Bachelier (1900). Théorie de la Spéculation, Ann. Sci. École Norm. Sup. 17, 21–86. [See also the 1995 reprint. Sceaux: Gauthier—Villars.]

    Google Scholar 

  2. V. V. Baklan (1965). The existence of solutions for a class of equations involving variational derivatives, Dopovidi Akad. Nauk Ukraĭn. RSR 1965, 554–556 (Ukranian. Russian, English summary)

    MATH  MathSciNet  Google Scholar 

  3. D. L. Burkholder (1971). Martingale inequalities, In: Lecture Notes in Math. 190, 1–8 Springer-Verlag, Berlin

    Google Scholar 

  4. E. M. Cabaña (1970). The vibrating string forced by white noise, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 15, 111–130

    Article  MATH  Google Scholar 

  5. Robert C. Dalang (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.', Electron. J. Probab. 4, no. 6, 29 pages (electronic)

    MathSciNet  Google Scholar 

  6. Robert C. Dalang, Davar Khoshnevisan, and Eulalia Nualart (2007). Hitting probabilities for systems of non-linear stochastic heat equations with additive noise, Latin American J. Probab. and Math. Statist. (or Alea; http://alea.impa.br/english), Vol. III, 231–371

    Google Scholar 

  7. Ju. L. Daleckiĭ(1967). Infinite-dimensional elliptic operators and the corresponding parabolic equations, Uspehi Mat. Nauk 22 (4) (136), 3–54 (In Russian) [English translation in: Russian Math. Surveys 22 (4), 1–53, 1967]

    Google Scholar 

  8. D. A. Dawson (1975). Stochastic evolution equations and related measure processes, J. Multivariate Anal. 5, 1–52

    Article  MATH  MathSciNet  Google Scholar 

  9. D. A. Dawson (1972). Stochastic evolution equations, Math. Biosci. 15, 287–316

    Article  MATH  MathSciNet  Google Scholar 

  10. J. L. Doob (1942). The Brownian movement and stochastic equations, Ann. of Math. 43 (2), 351–369

    Article  MathSciNet  Google Scholar 

  11. R. M. Dudley (1967). The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis, 1, 290–330

    Article  MATH  MathSciNet  Google Scholar 

  12. X. Fernique (1975). Regularité des trajectoires des fonctions aléatoires gaussiennes, In: Lecture Notes in Math. 480, 1–96 Springer-Verlag, Berlin (in French )

    Google Scholar 

  13. Tadahisa Funaki (1984). Random motion of strings and stochastic differential equations on the space C([0, 1], Rd), In: Stochastic Analysis (Katata/Kyoto, 1982), North-Holland Math. Library, 32, 121–133, North-Holland, Amsterdam

    Google Scholar 

  14. Tadahisa Funaki (1983). Random motion of strings and related stochastic evolution equations, Nagoya Math. J. 89, 129–193

    MATH  MathSciNet  Google Scholar 

  15. Kiyosi Itô (1944). Stochastic integral, Proc. Imp. Acad. Tokyo 20, 519–524

    Article  MATH  MathSciNet  Google Scholar 

  16. Kiyosi Itô (1950). Stochastic differential equations in a differentiable manifold, Nagoya Math. J. 1, 35–47

    MATH  MathSciNet  Google Scholar 

  17. Kiyosi Itô (1951). On a formula concerning stochastic differentials, Nagoya Math. J. 3, 55–65

    MATH  MathSciNet  Google Scholar 

  18. A. Ya. Khintchine (1933). Asymptotische Gesetz der Wahrscheinlichkeitsrechnung, Springer, Berlin

    Google Scholar 

  19. A. N. Kolmogorov (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung,Springer, Berlin

    Google Scholar 

  20. N. V. Krylov and B. L. Rozovskiĭ (1979a). Itô equations in Banach spaces and strongly parabolic stochastic partial differential equations, Dokl. Akad. Nauk SSSR 249 (2), 285–289 (in Russian)

    Google Scholar 

  21. N. V. Krylov and B. L. Rozovskiĭ (1979b). Stochastic evolution equations, In: Current Problems in Mathematics, Vol. 14 Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 71–147, 256 (in Russian)

    Google Scholar 

  22. N. V. Krylov, N. V. and B. L. Rozovskiĭ (1977). The Cauchy problem for linear stochastic partial differential equations, Izv. Akad. Nauk SSSR Ser. Mat. 41 (6),1329–1347, 1448 (in Russian)

    MATH  MathSciNet  Google Scholar 

  23. H. Kunita (1991). Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge

    Google Scholar 

  24. Étienne Pardoux (1975). Equations aux dérivées partielles stochastiques nonlinéaires monotones—Étude de solutions fortes de type Itô, Thése d'État, Univ. Paris XI, Orsay

    Google Scholar 

  25. Étienne Pardoux (1972). Sur des équations aux dérivées partielles stochastiques monotones, C. R. Acad. Sci. Paris Sér. A—B 275, A101–A103

    Google Scholar 

  26. Christopher Preston (1972). Continuity properties of some Gaussian processes,Ann. Math. Statist. 43, 285–292

    Article  MATH  MathSciNet  Google Scholar 

  27. Michel Talagrand (1985). Régularité des processus gaussiens, C. R. Acad. Sci. Paris Sér. I Math. 301 (7), 379–381 (French, with English summary)

    MATH  MathSciNet  Google Scholar 

  28. Michel Talagrand (1987). Regularity of Gaussian processes, Acta Math. 159 (1–2), 99–149

    Article  MathSciNet  Google Scholar 

  29. G. E. Uhlenbeck and L. S. Ornstein (1930). On the theory of Brownian Motion, Phys. Rev. 36, 823–841

    Article  Google Scholar 

  30. John B. Walsh (1986). An Introduction to Stochastic Partial Differential Equations, In: Lecture Notes in Math. 1180, 265–439, Springer, Berlin

    Google Scholar 

  31. N. Wiener (1923). Differential space, J. Math. Phys. 2, 131–174

    Google Scholar 

  32. Norbert Wiener (1938). The Homogeneous Chaos, Amer. J. Math. 60 (4), 897–936

    Article  MathSciNet  Google Scholar 

Download references

Authors
  1. Davar Khoshnevisan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Davar Khoshnevisan Firas Rassoul-Agha

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Khoshnevisan, D. (2009). A Primer on Stochastic Partial Differential Equations. In: Khoshnevisan, D., Rassoul-Agha, F. (eds) A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol 1962. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85994-9_1

Download citation

Publish with us

Policies and ethics

Search

Navigation