The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity

Marta Sanz-Solé (University of Barcelona)
André Süss (University of Barcelona)


We consider the class of non-linear stochastic partial differential equations studied in [Conus-Dalang, 2008]. Equivalent formulations using integration with respect to a cylindrical Brownian motion and also the Skorohod integral are established. It is proved that the random field solution to these equations at any fixed point $(t,x)\in[0,T]\times \mathbb{R}^d$ is differentiable in the Malliavin sense. For this, an extension of the integration theory in [Conus-Dalang, 2008] to Hilbert space valued integrands is developed, and commutation formulae of the Malliavin derivative and stochastic and pathwise integrals are proved. In the particular case of equations with additive noise, we establish the existence of density for the law of the solution at $(t,x)\in]0,T]\times\mathbb{R}^d$. The results apply to the stochastic wave equation in spatial dimension $d\ge 4$.

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Pages: 1-28

Publication Date: June 21, 2013

DOI: 10.1214/EJP.v18-2341


  • Bally, Vlad; Pardoux, Etienne. Malliavin calculus for white noise driven parabolic SPDEs. Potential Anal. 9 (1998), no. 1, 27--64. MR1644120
  • Carmona, René; Nualart, David. Random nonlinear wave equations: smoothness of the solutions. Probab. Theory Related Fields 79 (1988), no. 4, 469--508. MR0966173
  • Conus, Daniel; Dalang, Robert C. The non-linear stochastic wave equation in high dimensions. Electron. J. Probab. 13 (2008), no. 22, 629--670. MR2399293
  • Dalang, Robert C.; Frangos, N. E. The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 (1998), no. 1, 187--212. MR1617046
  • Dalang, Robert C. Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s. Electron. J. Probab. 4 (1999), no. 6, 29 pp. (electronic). MR1684157
  • Dalang, Robert C.; Quer-Sardanyons, Lluís. Stochastic integrals for spde's: a comparison. Expo. Math. 29 (2011), no. 1, 67--109. MR2785545
  • Da Prato, G. and Zabczyk, J.: Stochastic Equations in Infinite Dimensions. phCambridge University Press, Cambridge, 2008.
  • Kusuoka, Shigeo; Stroock, Daniel. Applications of the Malliavin calculus. I. Stochastic analysis (Katata/Kyoto, 1982), 271--306, North-Holland Math. Library, 32, North-Holland, Amsterdam, 1984. MR0780762
  • Lax, Peter D. Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002. xx+580 pp. ISBN: 0-471-55604-1 MR1892228
  • Márquez-Carreras, D.; Mellouk, M.; Sarrà, M. On stochastic partial differential equations with spatially correlated noise: smoothness of the law. Stochastic Process. Appl. 93 (2001), no. 2, 269--284. MR1828775
  • Millet, Annie; Sanz-Solé, Marta. A stochastic wave equation in two space dimension: smoothness of the law. Ann. Probab. 27 (1999), no. 2, 803--844. MR1698971
  • Nualart, D.; Sanz, M. Malliavin calculus for two-parameter Wiener functionals. Z. Wahrsch. Verw. Gebiete 70 (1985), no. 4, 573--590. MR0807338
  • Nualart, D.; Sanz, M. Stochastic differential equations on the plane: smoothness of the solution. J. Multivariate Anal. 31 (1989), no. 1, 1--29. MR1022349
  • Nualart, David. The Malliavin calculus and related topics. Second edition. Probability and its Applications (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp. ISBN: 978-3-540-28328-7; 3-540-28328-5 MR2200233
  • Nualart, David; Quer-Sardanyons, Lluís. Existence and smoothness of the density for spatially homogeneous SPDEs. Potential Anal. 27 (2007), no. 3, 281--299. MR2336301
  • Pardoux, Étienne; Zhang, Tu Sheng. Absolute continuity of the law of the solution of a parabolic SPDE. J. Funct. Anal. 112 (1993), no. 2, 447--458. MR1213146
  • Peszat, Szymon. The Cauchy problem for a nonlinear stochastic wave equation in any dimension. J. Evol. Equ. 2 (2002), no. 3, 383--394. MR1930613
  • Quer-Sardanyons, L.; Sanz-Solé, M. Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation. J. Funct. Anal. 206 (2004), no. 1, 1--32. MR2024344
  • Quer-Sardanyons, Lluís; Sanz-Solé, Marta. A stochastic wave equation in dimension 3: smoothness of the law. Bernoulli 10 (2004), no. 1, 165--186. MR2044597
  • Sanz-Solé, Marta. Malliavin calculus. With applications to stochastic partial differential equations. Fundamental Sciences. EPFL Press, Lausanne; distributed by CRC Press, Boca Raton, FL, 2005. viii+162 pp. ISBN: 2-940222-06-1; 0-8493-4030-6 MR2167213
  • Sanz-Solé, Marta. Properties of the density for a three-dimensional stochastic wave equation. J. Funct. Anal. 255 (2008), no. 1, 255--281. MR2417817
  • Schwartz, L.: Théorie des Distributions. phHermann, 2nd edition, Paris, 2010.
  • Trèves, François. Topological vector spaces, distributions and kernels. Unabridged republication of the 1967 original. Dover Publications, Inc., Mineola, NY, 2006. xvi+565 pp. ISBN: 0-486-45352-9 MR2296978
  • Walsh, John B. An introduction to stochastic partial differential equations. École d'été de probabilités de Saint-Flour, XIV—1984, 265--439, Lecture Notes in Math., 1180, Springer, Berlin, 1986. MR0876085

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