The stochastic wave equation in two spatial dimensions. (English) Zbl 0938.60046

Summary: We consider the wave equation in two spatial dimensions driven by space-time Gaussian noise that is white in time but has a nondegenerate spatial covariance. We give a necessary and sufficient integral condition on the covariance function of the noise for the solution to the linear form of the equation to be a real-valued stochastic process, rather than a distribution-valued random variable. When this condition is satisfied, we show that not only the linear form of the equation, but also nonlinear versions, have a real-valued process solution. We give stronger sufficient conditions on the spatial covariance for the solution of the linear equation to be continuous, and we provide an estimate of its modulus of continuity.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
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[1] Adams, R. A. (1975). Sobolev Spaces. Academic Press, New York. · Zbl 0314.46030
[2] Albeverio, S., Haba,and Russo, F. (1996). Trivial solutions for a non-linear two space dimensional wave equation perturbed by a space-time white noise. Stochastics Stochastics Rep. 56 127-160. · Zbl 0887.60069
[3] Albeverio, S. and Hoegh-Kron, A. (1984). Diffusion fields, quantum fields and fields with values in Lie groups. In Stochastic Analysis and Related Topics (M. A. Pinsky, ed.) 7 1-98. Dekker, New York. · Zbl 0575.60078
[4] Biswas, S. K. and Ahmed, N. U. (1985). Stabilisation of systems governed by the wave equation in the presence of distributed white noise. IEEE Trans. Automat. Control AC-30 1043-1045. · Zbl 0579.93065
[5] Caba na, E. (1972). On barrier problems for the vibrating string.Wahrsch. Verw. Gebiete 22 13-24. · Zbl 0214.16801
[6] Carmona, R. and Nualart, D. (1988). Random nonlinear wave equations: propagation of singularities. Ann. Probab. 16 730-751. · Zbl 0643.60045
[7] Dalang, R. C. and Walsh, J. B. (1992). The sharp Markov property of Lévy sheets. Ann. Probab. 20 591-626. · Zbl 0783.60049
[8] Dellacherie, C. and Meyer, P. A. (1975). Probabilités et Potentiel. Hermann, Paris. · Zbl 0323.60039
[9] Dunford, N. and Schwarz, J. T. (1988). Linear Operators, Part I: General Theory. Wiley, New York. · Zbl 0635.47001
[10] Léandre, R. and Russo, F. (1990). Estimation de Varadhan pour des diffusions a deux param etres. Probab. Theory Related Fields 84 429-451. · Zbl 0665.60057
[11] Léandre, R. and Russo, F. (1992). Small stochastic perturbation of a non-linear stochastic wave equation. In Progress in Probability (H. Korezlioglu and A. S. Ustunel, eds.) 31 285-332. Birkhäuser, Boston. · Zbl 0790.60053
[12] Marcus, M. and Mizel, V. J. (1991). Stochastic hyperbolic systems and the wave equation. Stochastics Stochastics Rep. 36 225-244. · Zbl 0739.60059
[13] Miller, R. N. (1990). Tropical data assimilation with simulated data: the impact of the tropical ocean and global atmosphere thermal array for the ocean. Journal of Geophysical Research 95 11,461-11,482.
[14] Mueller, C. (1997). Long time existence for the wave equation with a noise term. Ann. Probab. 25 133-151. · Zbl 0884.60054
[15] Nelson, E. (1973). The free Markoff field. J. Funct. Anal. 12 211-227. · Zbl 0273.60079
[16] Orsingher, E. (1982). Randomly forced vibrations of a string. Ann. Inst. H. Poincaré 18 367-394. · Zbl 0493.60067
[17] Stein, E. M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press. · Zbl 0207.13501
[18] Stroock, D. W. (1993). Probability Theory, an Analytic View. Cambridge Univ. Press. · Zbl 0925.60004
[19] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. Ecole d’Eté de Probabilités de Saint-Flour XIV. Lecture Notes in Math. 1180 265-439. Springer, Berlin. · Zbl 0608.60060
[20] Wheeden, R. L. and Zygmund, A. (1977). Measure and Integral: An Introduction to Real Analysis. Dekker, New York. · Zbl 0362.26004
[21] Wilcox, C. H. (1991). The Cauchy problem for the wave equation with distribution data: an elementary approach. Amer. Math. Monthly 98 401-410. JSTOR: · Zbl 0752.35032
[22] Yeh, J. (1981). Existence of strong solutions for stochastic differential equations in the plane. Pacific J. Math. 97 217-247. · Zbl 0516.60068
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