Skip to main content

Probability distributions of solutions to some stochastic partial differential equations

Part of the Lecture Notes in Mathematics book series (LNM,volume 1236)

Abstract

We consider a stochastic p.d.e. whose solution p(t) evolves in L2(ℝn). At each t, the probability distribution of p(t) is a measure on L2(ℝn). We define a Lie algebra naturally associated to the dynamics of p(t) from the operators of the stochastic p.d.e. and show that if this algebra, applied to the initial condition, ‘spans’ L2(ℝn), then the distribution of p(t) restricted to any cylinder set, is absolutely continuous with respect to Lebesgue measure on that cylinder set. Our motivation stems partly from issues in nonlinear filtering theory.

Keywords

  • Stochastic Calculus
  • Stochastic Partial Differential Equation
  • Wiener Space
  • Finite Dimensional Subspace
  • Wiener Measure

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Baras, G. L. Blankenship, W. E. Hopkins, Existence, Uniqueness, and Asymptocic Behavior of Solutions to a Class of Zakai Equations with Unbounded Coefficients, IEEE Trans. Automat. Control AC-28 (1983), 203–214.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. C. Bardos, L. Tartar, Sur l'Unicité des Equation Paraboliques et Quelques Questions Voisines, Archive for Rational Mechanics and Analysis 50 (1973), 10–25.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. J. M. Ball, J. E. Marsden, M. Slemrod, Controllability for Distributed Bilinear Systems, SIAM J. Control and Optimization 20 (1982), 575–597.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. A. Bensoussan, J-L. Lions, Applications of Variational Inequalities in Stochastic Control, North-Holland, Amsterdam (1982).

    MATH  Google Scholar 

  5. R. W. Brockett, J. M. C. Clark, On the geometry of the Conditional Density Equation, in Analysis and Optimisation of Stochastic Systems, ed. D. L. R. Jacobs, Academic Press (1980)

    Google Scholar 

  6. S. I. Marcus, Algebraic and Geometric Methods in Nonlinear Filtering, SIAM J. Control and Optimization 22 (1984), 817–844.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, Proc. Intern. Symp. S.D.E. Kyoto, ed. by K. Ito, Kinokuniya, Tokyo (1978).

    Google Scholar 

  8. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, 1964.

    Google Scholar 

  9. D. Ocone, Application of Wiener Space Analysis to Nonlinear Filtering, Proceedings of MTNS-85, Stockholm, June 1985, to appear, North-Holland press.

    Google Scholar 

  10. D. Ocone, Application of stochastic calculus of variation to stochastic partial differential equations, in preparation.

    Google Scholar 

  11. E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics 3 (1979), 127–167.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. G. Ferreyra, On the degenerate parabolic partial differential equations of nonlinear filtering, Comm. Partial Differential Equations 10 (1985), 555–634.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. I. Shigekawa, Derivatives of Wiener functionals and absolute continuity of induced measures, J. Math. Kyoto Univ. 20 (1980), 263–289.

    MathSciNet  MATH  Google Scholar 

  14. M. Hazewinkel, J. C. Willems, eds., Stochastic Systems; The Mathematics of Filtering and Identification and Applications, Reidel, Dordrecht (1981).

    Google Scholar 

  15. D. W. Stroock, Some Applications of Stochastic Calculus to Partial Differential Equations, in Ecole d'Eté de Probabilités de Saint-Flour XI-1981, Springer Lecture Notes 976, Springer-Verlag, Berlin (1983).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1987 Springer-Verlag

About this paper

Cite this paper

Ocone, D. (1987). Probability distributions of solutions to some stochastic partial differential equations. In: Da Prato, G., Tubaro, L. (eds) Stochastic Partial Differential Equations and Applications. Lecture Notes in Mathematics, vol 1236. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072890

Download citation

  • DOI: https://doi.org/10.1007/BFb0072890

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17211-6

  • Online ISBN: 978-3-540-47408-1

  • eBook Packages: Springer Book Archive