The Annals of Probability

Second-order linear hyperbolic SPDEs driven by isotropic Gaussian noise on a sphere

Robert C. Dalang and Olivier Lévêque

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Abstract

We study a class of linear hyperbolic stochastic partial differential equations in bounded domains, which includes the wave equation and the telegraph equation, driven by Gaussian noise that is white in time but not in space. We give necessary and sufficient conditions on the spatial correlation of the noise for the existence (and uniqueness) of square-integrable solutions. In the particular case where the domain is a ball and the noise is concentrated on a sphere, we characterize the isotropic Gaussian noises with this property. We also give explicit necessary and sufficient conditions when the domain is a hypercube and the Gaussian noise is concentrated on a hyperplane.

Article information

Source
Ann. Probab., Volume 32, Number 1B (2004), 1068-1099.

Dates
First available in Project Euclid: 11 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.aop/1079021472

Digital Object Identifier
doi:10.1214/aop/1079021472

Mathematical Reviews number (MathSciNet)
MR2044674

Zentralblatt MATH identifier
1046.60058

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60G15: Gaussian processes 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Keywords
Stochastic partial differential equations isotropic Gaussian noise hyperbolic equations

Citation

Dalang, Robert C.; Lévêque, Olivier. Second-order linear hyperbolic SPDEs driven by isotropic Gaussian noise on a sphere. Ann. Probab. 32 (2004), no. 1B, 1068--1099. doi:10.1214/aop/1079021472. https://projecteuclid.org/euclid.aop/1079021472


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References

  • Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC.
    Zentralblatt MATH: 0171.38503
    Mathematical Reviews (MathSciNet): MR167642
  • Adams, R. A. (1975). Sobolev Spaces. Academic Press, New York.
    Mathematical Reviews (MathSciNet): MR450957
    Zentralblatt MATH: 0314.46030
  • Alòs, E. and Bonnacorsi, S. (2002). Stochastic partial differential equations with Dirichlet white-noise boundary conditions. Ann. Inst. H. Poincaré Probab. Statist. 38 125--154.
    Mathematical Reviews (MathSciNet): MR1899108
    Digital Object Identifier: doi:10.1016/S0246-0203(01)01097-4
  • Breen, S. (1995). Uniform upper and lower bounds on the zeros of Bessel functions of the first kind. J. Math. Anal. Appl. 196 1--17.
    Mathematical Reviews (MathSciNet): MR1359929
    Digital Object Identifier: doi:10.1006/jmaa.1995.1395
    Zentralblatt MATH: 0845.33002
  • Dalang, R. C. (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s. Electron. J. Probab. 4 1--29.
    Mathematical Reviews (MathSciNet): MR1684157
  • Dalang, R. C. (2001). Corrections to extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s. Electron. J. Probab. 6 1--5.
    Mathematical Reviews (MathSciNet): MR1825714
  • Dalang, R. C. and Frangos, N. E. (1998). The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 187--212.
    Mathematical Reviews (MathSciNet): MR1617046
    Digital Object Identifier: doi:10.1214/aop/1022855416
    Project Euclid: euclid.aop/1022855416
    Zentralblatt MATH: 0938.60046
  • Dalang, R. C. and Lévêque, O. (2003). Second-order hyperbolic s.p.d.e.'s driven by homogeneous Gaussian noise on a hyperplane. Preprint.
  • Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press.
    Mathematical Reviews (MathSciNet): MR1207136
  • Da Prato, G. and Zabczyk, J. (1993). Evolution equations with white-noise boundary conditions. Stochastics Stochastics Rep. 42 167--182.
    Mathematical Reviews (MathSciNet): MR1291187
  • Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. Cambridge Univ. Press.
    Mathematical Reviews (MathSciNet): MR1417491
    Zentralblatt MATH: 0849.60052
  • Dawson, D. and Salehi, H. (1980). Spatially homogeneous random evolutions. J. Multivariate Anal. 10 141--180.
    Mathematical Reviews (MathSciNet): MR575923
    Digital Object Identifier: doi:10.1016/0047-259X(80)90012-3
    Zentralblatt MATH: 0439.60051
  • Gradshteyn, I. S. and Ryshik, I. M. (1994). Table of Integrals, Series and Products. Academic Press, New York.
    Mathematical Reviews (MathSciNet): MR1243179
    Zentralblatt MATH: 0918.65002
  • Karczewska, A. and Zabczyk, J. (1999). Stochastic PDE's with function-valued solutions. In Infinite-Dimensional Stochastic Analysis. (Ph. Clément, F. den Hollander, T. van Neerven and B. de Pagter, eds.) 197--216. Royal Netherlands Academy of Arts and Sciences, Amsterdam.
    Mathematical Reviews (MathSciNet): MR1832378
  • Malliavin, P. (1993). Integration and Probability. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1335234
  • Mao, X. and Markus, L. (1993). Wave equation with stochastic boundary values. J. Math. Anal. Appl. 177 315--341.
    Mathematical Reviews (MathSciNet): MR1231485
    Digital Object Identifier: doi:10.1006/jmaa.1993.1261
    Zentralblatt MATH: 0784.60061
  • Maslowski, B. (1995). Stability of semilinear equations with boundary and pointwise noise. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 55--93.
    Mathematical Reviews (MathSciNet): MR1315350
  • Miller, R. N. (1990). Tropical data assimilation with simulated data: The impact of the tropical ocean and global atmosphere thermal array for the ocean. J. Geophys. Res. 95 11461--11482.
  • Millet, A. and Sanz-Solé, M. (1999). A stochastic wave equation in two space dimension: Smoothness of the law. Ann. Probab. 27 803--844.
    Mathematical Reviews (MathSciNet): MR1698971
    Digital Object Identifier: doi:10.1214/aop/1022677387
    Project Euclid: euclid.aop/1022677387
  • Müller, C. (1998). Analysis of Spherical Symmetries in Euclidean Spaces. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1483320
  • Neveu, J. (1968). Processus aléatoires gaussiens. Univ. Montreal Press.
    Mathematical Reviews (MathSciNet): MR272042
  • Oberguggenberger, M. and Russo, F. (1997). The non-linear stochastic wave equation. Integral Transforms and Special Functions 6 58--70.
  • Peszat, S. (2002). The Cauchy problem for a nonlinear stochastic wave equation in any dimension. Journal of Evolution Equations 2 383--394.
    Mathematical Reviews (MathSciNet): MR1930613
    Digital Object Identifier: doi:10.1007/PL00013197
  • Peszat, S. and Zabczyk, J. (1997). Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Process. Appl. 72 187--204.
    Mathematical Reviews (MathSciNet): MR1486552
    Digital Object Identifier: doi:10.1016/S0304-4149(97)00089-6
    Zentralblatt MATH: 0943.60048
  • Peszat, S. and Zabczyk, J. (2000). Nonlinear stochastic wave and heat equations. Probab. Theory Related Fields 116 421--443.
    Mathematical Reviews (MathSciNet): MR1749283
    Digital Object Identifier: doi:10.1007/s004400050257
    Zentralblatt MATH: 0959.60044
  • Protter, Ph. (1990). Stochastic Integration and Differential Equations. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1037262
    Zentralblatt MATH: 0694.60047
  • Schönberg, I. J. (1942). Positive definite functions on spheres. Duke Math. J. 9 96--108.
    Mathematical Reviews (MathSciNet): MR5922
    Digital Object Identifier: doi:10.1215/S0012-7094-42-00908-6
    Project Euclid: euclid.dmj/1077493072
    Zentralblatt MATH: 0063.06808
  • Schwartz, L. (1966). Théorie des distributions. Hermann, Paris.
    Mathematical Reviews (MathSciNet): MR209834
  • Shimakura, N. (1992). Partial Differential Operators of Elliptic Type. Amer. Math. Soc., Providence, RI.
    Mathematical Reviews (MathSciNet): MR1168472
    Zentralblatt MATH: 0757.35015
  • Sowers, R. (1994). Multidimensional reaction--diffusion equations with white noise boundary perturbations. Ann. Probab. 22 2071--2121.
    Mathematical Reviews (MathSciNet): MR1331216
    JSTOR: links.jstor.org
  • Walsh, J. B. (1986). An introduction to stochastic partial differential equations. Lecture Notes in Math. 1180 266--439. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR876085
    Zentralblatt MATH: 0608.60060

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