Potential Analysis

, Volume 38, Issue 2, pp 345–379 | Cite as

Weak Order for the Discretization of the Stochastic Heat Equation Driven by Impulsive Noise

  • Felix LindnerEmail author
  • René L. Schilling


We study the approximation of the distribution of X T , where (X t ) t ∈ [0, T] is a Hilbert space valued stochastic process that solves a linear parabolic stochastic partial differential equation driven by an impulsive space time noise,
$$ dX_t+AX_t\,dt= Q^{1/2}\,dZ_t,\quad X_0=x_0\in H,\quad t\in [0,T]. $$
Here (Z t ) t ∈ [0, T] is an impulsive cylindrical process and the operator Q describes the spatial covariance structure of the noise; we assume that A − α has finite trace for some α > 0 and that A β Q is bounded for some β ∈ (α − 1, α]. A discretized solution \((X_h^n)_{n\in\{0,1,\ldots,N\}}\) is defined via the finite element method in space (parameter h > 0) and a θ-method in time (parameter Δt = T/N). For \(\varphi \in C^2_b(H;{\mathbb R})\) we show an integral representation for the error \(|{\mathbb E}\varphi(X^N_h)-{\mathbb E}\varphi(X_T)|\) and prove that
$$ \left|{\mathbb E}\varphi\left(X^N_h\right)-{\mathbb E}\varphi(X_T)\right|=O\left(h^{2\gamma}+\left(\Delta t\right)^{\gamma}\right) $$
where γ < 1 − α + β. This is the same order of convergence as in the case of a Gaussian space time noise, which has been obtained in a paper by Debussche and Printems (Math Comput 78:845–863, 2009). Our result also holds for a combination of impulsive and Gaussian space time noise.


Weak order Stochastic heat equation Impulsive cylindrical process Infinite dimensional Lévy process Finite element Euler scheme 

Mathematics Subject Classifications (2000)

Primary 60H15 65M60; Secondary 60H35 60G51 60G52 65C30 


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  1. 1.
    Applebaum, D., Wu, J.L.: Stochastic partial differential equations driven by Lévy space-time white noise. Random Oper. Stoch. Equ. 3, 245–261 (2000)MathSciNetGoogle Scholar
  2. 2.
    Bramble, J.H., Schatz, A.H., Thomée, V., Wahlbin, L.B.: Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations. SIAM J. Numer. Anal. 14, 218–241 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Bruti-Liberati, N., Platen, E.: Approximation of jump diffusions in finance and economics. Comput. Econ. 29, 283–312 (2007)zbMATHCrossRefGoogle Scholar
  4. 4.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)zbMATHGoogle Scholar
  5. 5.
    Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. In: Encyclopedia of Mathematics and Its Applications, vol. 44. Cambridge University Press, Cambridge (1992)Google Scholar
  6. 6.
    de Bouard, A., Debussche, A.: Weak and strong order of convergence of a semidiscrete scheme for the stochastic nonlinear Schrödinger equation. Appl. Math. Optim. 54, 369–399 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Debussche, A.: Weak approximation of stochastic partial differential equations: the non linear case. Math. Comput. 80, 89–117 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Debussche, A., Printems, J.: Weak order for the discretization of the stochastic heat equation. Math. Comput. 78, 845–863 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Geissert, M., Kovács, M., Larsson, S.: Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise. BIT 49, 343–356 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gyöngy, I., Millet, A.: On discretization schemes for stochastic evolution equations. Potential Anal. 23, 99–134 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Gyöngy, I., Millet, A.: Rate of convergence of implicit approximations for stochastic evolution equations. In: Baxendale, P.H., Lototsky, S.V. (eds.) Stochastic Differential Equations: Theory and Applications. A Volume in Honor of Professor Boris L. Rozovskii. Interdisciplinary Mathematical Siences, vol. 2, pp. 281–310. World Scientific, Hackensack (2007)CrossRefGoogle Scholar
  12. 12.
    Gyöngy, I., Millet, A.: Rate of convergence of space time approximations for stochastic evolution equations. Potential Anal. 30, 29–64 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hausenblas, E.: Weak approximation for semiliniear stochastic evolution equations. In: Capar, U., Üstünel, A. (eds.) Stochastic Analysis and Related Topics VIII. Silivri Workshop. Progress in Probability, pp. 111–128. Birkhäuser, Basel (2003)Google Scholar
  14. 14.
    Hausenblas, E.: Existence, uniqueness and regularity of parabolic SPDEs driven by Poisson random measure. Electron. J. Probab. 10, 1496–1546 (2005)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Hausenblas, E.: Finite element approximation of stochastic partial differential equations driven by Poisson random measures of jump type. SIAM J. Numer. Anal. 46, 437–471 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hausenblas, E.: Weak approximation of the stochastic wave equation. J. Comput. Appl. Math. 235(1), 33–58 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Hausenblas, E., Marchis, I.: A numerical approximation of parabolic stochastic partial differential equations driven by a Poisson random measure. BIT 46, 773–811 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Johnson, C., Larsson, S., Thomée, V., Wahlbin, B.: Error estimates for spatially discrete approximations of semilinear parabolic equations with non smooth initial data. Math. Comput. 49, 331–357 (1987)zbMATHCrossRefGoogle Scholar
  19. 19.
    Knoche, C.: SPDEs in infinite dimension with Poisson noise. C. R. Math. Acad. Sci. Paris, Ser. I 339, 647–652 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Le Roux, M.-N.: Semidiscretization in time for parabolic problems. Math. Comput. 33, 919–931 (1979)zbMATHGoogle Scholar
  21. 21.
    Liu, X.Q., Li, C.W.: Weak approximations and extrapolations of stochastic differential equations with jumps. SIAM J. Numer. Anal. 37, 1747–1767 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Marinelli, C., Prévot, C., Röckner, M.: Regular dependence on initial data for stochastic convolution equations with multiplicative Poisson noise. J. Funct. Anal. 258, 616–649 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Métivier, M.: Semimartingales—a course on stochastic processes. In: de Gruyter Studies in Mathematics, vol. 2. de Gruyter, Berlin (1982)Google Scholar
  24. 24.
    Métivier, M., Pellaumail, J.: Stochastic Integration. Probability and Mathematical Statistics. Academic Press, New York (1980)zbMATHGoogle Scholar
  25. 25.
    Miculevičius, R., Platen, E.: Time discrete Taylor approximations for Itô processes with jump component. Math. Nachr. 138, 93–104 (1988)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mueller, C.: The heat equation with Lévy noise. Stoch. Process. Their Appl. 74, 67–82 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Mytnik, L.: Stochastic partial differential equation driven by stable noise. Probab. Theory Relat. Fields 123, 157–201 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Peszat, S., Zabczyk, J.: Stochastic partial differential equations with Lévy noise. In: Encyclopedia of Mathematics and Its Applications, vol. 113. Cambridge University Press, Cambridge (2007)Google Scholar
  29. 29.
    Protter, P., Talay, D.: The Euler scheme for Lévy driven stochastic differential equations. Ann. Probab. 25, 393–423 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Rozanov, Yu.A.: Random Fields and Stochastic Partial Differential Equations. Kluwer Academic, Dordrecht (1998)zbMATHGoogle Scholar
  31. 31.
    Sato, K.: Lévy processes and infinitely divisible distributions. In: Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge (2005)Google Scholar
  32. 32.
    Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Wellesley Cambridge Press, Cambridge (1973)zbMATHGoogle Scholar
  33. 33.
    Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer, Berlin (1997)zbMATHGoogle Scholar
  34. 34.
    Walsh, J.B.: An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour XIV-1984. Lecture Notes in Mathematics, vol. 1180, pp. 265–439. Springer, Berlin (1986)CrossRefGoogle Scholar
  35. 35.
    Walsh, J.B.: Finite element methods for parabolic stochastic PDEs. Potential Anal. 23, 1–43 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Yan, Y.: Galerkin finite element methods for stochastic partial differential equations. SIAM J. Numer. Anal. 43, 1363–1384 (2005)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Institut für Mathematische StochastikTechnische Universität DresdenDresdenGermany

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