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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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

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

First available in Project Euclid: 11 March 2004

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Zentralblatt MATH identifier

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]

Stochastic partial differential equations isotropic Gaussian noise hyperbolic equations


Dalang, Robert C.; Lévêque, Olivier. Second-order linear hyperbolic SPDEs driven by isotropic Gaussian noise on a sphere. The Annals of Probability 32 (2004), no. 1B, 1068--1099. doi:10.1214/aop/1079021472.

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  • Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC.
  • Adams, R. A. (1975). Sobolev Spaces. Academic Press, New York.
  • 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.
  • 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.
  • 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.
  • 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.
  • Dalang, R. C. and Frangos, N. E. (1998). The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 187--212.
  • 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.
  • Da Prato, G. and Zabczyk, J. (1993). Evolution equations with white-noise boundary conditions. Stochastics Stochastics Rep. 42 167--182.
  • Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. Cambridge Univ. Press.
  • Dawson, D. and Salehi, H. (1980). Spatially homogeneous random evolutions. J. Multivariate Anal. 10 141--180.
  • Gradshteyn, I. S. and Ryshik, I. M. (1994). Table of Integrals, Series and Products. Academic Press, New York.
  • 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.
  • Malliavin, P. (1993). Integration and Probability. Springer, New York.
  • Mao, X. and Markus, L. (1993). Wave equation with stochastic boundary values. J. Math. Anal. Appl. 177 315--341.
  • Maslowski, B. (1995). Stability of semilinear equations with boundary and pointwise noise. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 55--93.
  • 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.
  • Müller, C. (1998). Analysis of Spherical Symmetries in Euclidean Spaces. Springer, New York.
  • Neveu, J. (1968). Processus aléatoires gaussiens. Univ. Montreal Press.
  • 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.
  • Peszat, S. and Zabczyk, J. (1997). Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Process. Appl. 72 187--204.
  • Peszat, S. and Zabczyk, J. (2000). Nonlinear stochastic wave and heat equations. Probab. Theory Related Fields 116 421--443.
  • Protter, Ph. (1990). Stochastic Integration and Differential Equations. Springer, Berlin.
  • Schönberg, I. J. (1942). Positive definite functions on spheres. Duke Math. J. 9 96--108.
  • Schwartz, L. (1966). Théorie des distributions. Hermann, Paris.
  • Shimakura, N. (1992). Partial Differential Operators of Elliptic Type. Amer. Math. Soc., Providence, RI.
  • Sowers, R. (1994). Multidimensional reaction--diffusion equations with white noise boundary perturbations. Ann. Probab. 22 2071--2121.
  • Walsh, J. B. (1986). An introduction to stochastic partial differential equations. Lecture Notes in Math. 1180 266--439. Springer, Berlin.