×

zbMATH — the first resource for mathematics

Second-order linear hyperbolic SPDEs driven by isotropic Gaussian noise on a sphere. (English) Zbl 1046.60058
The authors consider linear hyperbolic stochastic partial differential equations in bounded domains, driven by Gaussian noise that is white in time but not in space. The wave equation and the telegraph equation are immediate particular cases.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC. · Zbl 0171.38503
[2] Adams, R. A. (1975). Sobolev Spaces. Academic Press, New York. · Zbl 0314.46030
[3] 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. · Zbl 0998.60065
[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. · Zbl 0845.33002
[5] 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. · Zbl 0922.60056
[6] 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. · Zbl 0986.60053
[7] Dalang, R. C. and Frangos, N. E. (1998). The stochastic wave equation in two spatial dimensions. Ann. Probab. 26 187–212. · Zbl 0938.60046
[8] Dalang, R. C. and Lévêque, O. (2003). Second-order hyperbolic s.p.d.e.’s driven by homogeneous Gaussian noise on a hyperplane. · Zbl 1088.60065
[9] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44 . Cambridge Univ. Press. · Zbl 0761.60052
[10] Da Prato, G. and Zabczyk, J. (1993). Evolution equations with white-noise boundary conditions. Stochastics Stochastics Rep. 42 167–182. · Zbl 0814.60055
[11] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. Cambridge Univ. Press. · Zbl 0849.60052
[12] Dawson, D. and Salehi, H. (1980). Spatially homogeneous random evolutions. J. Multivariate Anal. 10 141–180. · Zbl 0439.60051
[13] Gradshteyn, I. S. and Ryshik, I. M. (1994). Table of Integrals , Series and Products. Academic Press, New York. · Zbl 0918.65002
[14] 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. · Zbl 0990.60065
[15] Malliavin, P. (1993). Integration and Probability. Springer, New York. · Zbl 0803.58009
[16] Mao, X. and Markus, L. (1993). Wave equation with stochastic boundary values. J. Math. Anal. Appl. 177 315–341. · Zbl 0784.60061
[17] Maslowski, B. (1995). Stability of semilinear equations with boundary and pointwise noise. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 55–93. · Zbl 0830.60056
[18] 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.
[19] Millet, A. and Sanz-Solé, M. (1999). A stochastic wave equation in two space dimension: Smoothness of the law. Ann. Probab. 27 803–844. · Zbl 0944.60067
[20] Müller, C. (1998). Analysis of Spherical Symmetries in Euclidean Spaces. Springer, New York. · Zbl 0884.33001
[21] Neveu, J. (1968). Processus aléatoires gaussiens. Univ. Montreal Press. · Zbl 0192.54701
[22] Oberguggenberger, M. and Russo, F. (1997). The non-linear stochastic wave equation. Integral Transforms and Special Functions 6 58–70.
[23] Peszat, S. (2002). The Cauchy problem for a nonlinear stochastic wave equation in any dimension. Journal of Evolution Equations 2 383–394. · Zbl 1375.60109
[24] Peszat, S. and Zabczyk, J. (1997). Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Process. Appl. 72 187–204. · Zbl 0943.60048
[25] Peszat, S. and Zabczyk, J. (2000). Nonlinear stochastic wave and heat equations. Probab. Theory Related Fields 116 421–443. · Zbl 0959.60044
[26] Protter, Ph. (1990). Stochastic Integration and Differential Equations. Springer, Berlin. · Zbl 0694.60047
[27] Schönberg, I. J. (1942). Positive definite functions on spheres. Duke Math. J. 9 96–108. · Zbl 0063.06808
[28] Schwartz, L. (1966). Théorie des distributions. Hermann, Paris.
[29] Shimakura, N. (1992). Partial Differential Operators of Elliptic Type . Amer. Math. Soc., Providence, RI. · Zbl 0757.35015
[30] Sowers, R. (1994). Multidimensional reaction–diffusion equations with white noise boundary perturbations. Ann. Probab. 22 2071–2121. JSTOR: · Zbl 0834.60067
[31] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. Lecture Notes in Math. 1180 266–439. Springer, Berlin. · Zbl 0608.60060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.