Electronic Journal of Probability

Some Non-Linear S.P.D.E's That Are Second Order In Time

Robert Dalang and Carl Mueller

Full-text: Open access

Abstract

We extend J.B. Walsh's theory of martingale measures in order to deal with stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert-Schmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms.

Article information

Source
Electron. J. Probab. Volume 8 (2003), paper no. 1, 21 p.

Dates
First available in Project Euclid: 23 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464037574

Digital Object Identifier
doi:10.1214/EJP.v8-123

Mathematical Reviews number (MathSciNet)
MR1961163

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 35L05: Wave equation

Keywords
Stochastic wave equation stochastic beam equation spatially homogeneous Gaussian noise stochastic partial differential equations

Citation

Dalang, Robert; Mueller, Carl. Some Non-Linear S.P.D.E's That Are Second Order In Time. Electron. J. Probab. 8 (2003), paper no. 1, 21 p. doi:10.1214/EJP.v8-123. https://projecteuclid.org/euclid.ejp/1464037574.


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References

  • Adams, R.A. Sobolev Spaces. Pure and Applied Mathematics, Vol. 65, Academic Press, New York-London, 1975. 
  • Carmona, R. and Nualart, D. Random nonlinear wave equations: propagation of singularities. Annals Probab. 16 (1988), 730-751. 
  • Carmona, R. and Nualart, D. Random nonlinear wave equations: smoothness of the solutions. Probab. Theory Related Fields 79 (1988), 469–508. 
  • Da Prato, G. and Zabczyk, J. Stochastic Equations in Infinite Dimensions. Encyclopedia of mathematics and its applications 44. Cambridge University Press, Cambridge, New York, 1992. 
  • Dalang, R.C. Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e's. Electron. J. Probab. 4 (1999), 29pp. 
  • Dalang, R.C. and Frangos, N.E. The stochastic wave equation in two spatial dimensions. Ann. Probab. 26-1 (1998), 187-212. 
  • Karczewska, A. and Zabczyk, J. Stochastic PDEs with function-valued solutions. In: Infinite dimensional stochastic analysis (Amsterdam, 1999), Royal Neth. Acad. Arts Sci. 52, Amsterdam (2000), 197-216. 
  • Krylov, N.V. and Rozovskii, B.L. Stochastic evolution systems. J. Soviet Math. 16 (1981), 1233-1276. 
  • Krylov, N.V. and Rozovskii, B.L. Stochastic partial differential equations and diffusion processes. Russian Math. Surveys 37 (1982), 81-105. 
  • Léveque, O. Hyperbolic stochastic partial differential equations driven by boundary noises. Ph.D. thesis, no.2452, Ecole Polytechnique Fédérale de Lausanne, Switzerland (2001). 
  • Millet, A. and Sanz-Solé, M. A stochastic wave equation in two space dimension: Smoothness of the law. Annals of Probab. 27 (1999), 803-844. 
  • Mueller, C. Long time existence for the wave equation with a noise term. Ann. Probab. 25-1 (1997), 133-152. 
  • Oberguggenberger, M. and Russo, F. White noise driven stochastic partial differential equations: triviality and non-triviality. In: Nonlinear Theory of Generalized Functions (M. Grosser, G. Hormann, M. Kunzinger & M. Oberguggenberger, eds), Chapman & Hall/CRC Research Notes in Mathematics Series, CRC Press (1999), 315-333.
  • Oberguggenberger, M. and Russo, F. Nonlinear stochastic wave equations. Integral Transform. Spec. Funct. 6 (1998), 71-83. 
  • Pardoux, E. Sur des équations aux dérivées partielles stochastiques monotones. C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A101-A103. 
  • Pardoux, E. Equations aux dérivées partielles stochastiques de type monotone. Séminaire sur les Equations aux Dérivées Partielles (1974–1975), III, Exp. No. 2 (1975), p.10. 
  • Pardoux, E. Characterization of the density of the conditional law in the filtering of a diffusion with boundary. In: Recent developments in statistics (Proc. European Meeting Statisticians, Grenoble, 1976). North Holland, Amsterdam (1977), 559-565. 
  • Peszat, S. The Cauchy problem for a nonlinear stochastic wave equation in any dimension. J. Evol. Equ. 2 (2002), 383-394.
  • Peszat, S. and Zabczyk, J. Stochastic evolution equations with a spatially homogeneous Wiener process. Stoch. Proc. Appl. 72 (1997), 187-204. 
  • Peszat, S. and Zabczyk, J. Nonlinear stochastic wave and heat equations. Probab. Theory Related Fields 116 (2000), 421-443. 
  • Sanz-Solé, M. and Sarra, M. Path properties of a class of Gaussian processes with applications to spde's. In: Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999). (Gestesy, F., Holden, H., Jost, J., Paycha, S., Rockner, M. and Scarlatti, S., eds). CMS Conf. Proc. 28, Amer. Math. Soc., Providence, RI (2000), 303-316. 
  • Schwartz, L. Théorie des distributions. Hermann, Paris (1966). 
  • Stein, E.M. Singular Integrals and Differentiability Properties of Functions. Princeton University Press (1970). 
  • Walsh, J.B. An introduction to stochastic partial differential equations, Ecole d'Eté de Prob. de St. Flour XIV, 1984, Lect. Notes in Math 1180, Springer-Verlag (1986).