Potential Analysis

, Volume 28, Issue 2, pp 139–162 | Cite as

Probabilistic Representations of Solutions of the Forward Equations

  • B. RajeevEmail author
  • S. Thangavelu


In this paper we prove a stochastic representation for solutions of the evolution equation
$$\partial _t \psi _t = \frac{1}{2}L^ * \psi _t $$
where L  ∗  is the formal adjoint of a second order elliptic differential operator L, with smooth coefficients, corresponding to the infinitesimal generator of a finite dimensional diffusion (X t ). Given ψ 0 = ψ, a distribution with compact support, this representation has the form ψ t  = E(Y t (ψ)) where the process (Y t (ψ)) is the solution of a stochastic partial differential equation connected with the stochastic differential equation for (X t ) via Ito’s formula.


Stochastic differential equation Stochastic partial differential equation Evolution equation Stochastic flows Ito’s formula Stochastic representation Adjoints Diffusion processes Second order elliptic partial differential equation Monotonicity inequality 

Mathematics Subject Classifications (2000)

Primary 60H10 60H15 Secondary 60J60 35K15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bass, R.F.: Diffusions and Elliptic Operators. Springer, New York (1997)Google Scholar
  2. 2.
    Bismut, J.M.: Mécanique aléatoire. In: Tenth Saint Flour Probability Summer School – 1980 (Saint Flour 1980) pp. 1–100. Springer, Berlin (1982)Google Scholar
  3. 3.
    Bensoussan, A., Lions, J.L.: Applications of Variational Inequalities in Stochastic Control. North Holland (1982)Google Scholar
  4. 4.
    Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications (1992)Google Scholar
  5. 5.
    Dynkin, E.B.: Markov Processes, Vols. 1–2. Springer, Berlin (1965)Google Scholar
  6. 6.
    Friedman, A.: Stochastic Differential Equations and Applications (Vol. 1). Academic (1975)Google Scholar
  7. 7.
    Gawarecki, L., Mandrekar, V., Rajeev, B.: Stochastic differential equations in the dual of a multi-Hilbertian space (2006) (pre-print)Google Scholar
  8. 8.
    Gawarecki, L., Mandrekar, V., Rajeev, B.: The monotonicity inequality for linear stochastic partial differential equations (2006) (pre-print)Google Scholar
  9. 9.
    Ito, K.: Foundations of stochastic differential equations in infinite dimensional spaces. Proceedings of CBMS-NSF National Conference in Applied Mathematics, SIAM (1982)Google Scholar
  10. 10.
    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North Holland (1981)Google Scholar
  11. 11.
    Kallianpur, G., Mitoma, I., Wolpert, R.L.: Diffusion equations in duals of nuclear spaces. Stoch. Reports 29, 285–329 (1990)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Krylov, N.V., Rozovskii, B.L.: Stochastic evolution equations. Itogi Nauki i Techniki 14, 71–146 (1981) English trans. by Plenum, pp. 1233–1277 (1981)Google Scholar
  13. 13.
    Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press (1990)Google Scholar
  14. 14.
    Métivier, M.: Semi-martingales – a course in Stochastic Processes. Walter de Gruyter (1982)Google Scholar
  15. 15.
    Pardoux, E.: Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3, 127–167 (1979)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Rajeev, B.: From Tanaka formula to Ito formula: distributions, tensor products and local Times. Seminaire de Probabilities XXXV, Lecture notes in Math. 1755, 371–389. Springer, Berlin (2001)Google Scholar
  17. 17.
    Rajeev, B., Thangavelu, S.: Probabilistic representations of solutions to the heat equation. Proc. Indian Acad. Sci. Math. Sci. 113(3), pp. 321–332 (2003)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Rozovskii, B.: Stochastic Evolution Systems: Linear Theory and Applications to Non Linear Filtering. Kluver Academic Publishers, Boston (1983)Google Scholar
  19. 19.
    Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1979)zbMATHGoogle Scholar
  20. 20.
    Thangavelu, S.: Lectures on Hermite and Laguerre Expansions. Math. Notes 42, Princeton University Press, Princeton (1993)Google Scholar
  21. 21.
    Treves, F.: Topological Vector Spaces, Distributions and Kernels. Academic, New York (1967)zbMATHGoogle Scholar
  22. 22.
    Walsh, J.B.: An Introduction to Stochastic Partial Differential Equations. Lecture notes in Mathematics 1180, Springer (1986)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Statistics and Mathematics UnitIndian Statistical InstituteBangaloreIndia
  2. 2.Department of MathematicsIndian Institute of ScienceBangaloreIndia

Personalised recommendations