Second-order hyperbolic s.p.d.e.’s driven by homogeneous Gaussian noise on a hyperplane
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- by Robert C. Dalang and Olivier Lévêque PDF
- Trans. Amer. Math. Soc. 358 (2006), 2123-2159 Request permission
Abstract:
We study a class of hyperbolic stochastic partial differential equations in Euclidean space, that includes the wave equation and the telegraph equation, driven by Gaussian noise concentrated on a hyperplane. The noise is assumed to be white in time but spatially homogeneous within the hyperplane. Two natural notions of solutions are function-valued solutions and random field solutions. For the linear form of the equations, we identify the necessary and sufficient condition on the spectral measure of the spatial covariance for existence of each type of solution, and it turns out that the conditions differ. In spatial dimensions 2 and 3, under the condition for existence of a random field solution to the linear form of the equation, we prove existence and uniqueness of a random field solution to non-linear forms of the equation.References
- Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
- Elisa Alòs and Stefano Bonaccorsi, Stochastic partial differential equations with Dirichlet white-noise boundary conditions, Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), no. 2, 125–154 (English, with English and French summaries). MR 1899108, DOI 10.1016/S0246-0203(01)01097-4
- Fritz Oberhettinger and Larry Badii, Tables of Laplace transforms, Springer-Verlag, New York-Heidelberg, 1973. MR 0352889
- Christian Berg, Jens Peter Reus Christensen, and Paul Ressel, Harmonic analysis on semigroups, Graduate Texts in Mathematics, vol. 100, Springer-Verlag, New York, 1984. Theory of positive definite and related functions. MR 747302, DOI 10.1007/978-1-4612-1128-0
- Robert C. Dalang, Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s, Electron. J. Probab. 4 (1999), no. 6, 29. MR 1684157, DOI 10.1214/EJP.v4-43
- Robert C. Dalang, Corrections to: “Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s” [Electron J. Probab. 4 (1999), no. 6, 29 pp. (electronic); MR1684157 (2000b:60132)], Electron. J. Probab. 6 (2001), no. 6, 5. MR 1825714
- Robert C. Dalang and N. E. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab. 26 (1998), no. 1, 187–212. MR 1617046, DOI 10.1214/aop/1022855416
- Donald A. Dawson and Habib Salehi, Spatially homogeneous random evolutions, J. Multivariate Anal. 10 (1980), no. 2, 141–180. MR 575923, DOI 10.1016/0047-259X(80)90012-3
- Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. MR 1207136, DOI 10.1017/CBO9780511666223
- G. Da Prato and J. Zabczyk, Evolution equations with white-noise boundary conditions, Stochastics Stochastics Rep. 42 (1993), no. 3-4, 167–182. MR 1291187, DOI 10.1080/17442509308833817
- Denis de Brucq and Christian Olivier, Approximations des processus gaussiens stationnaires, solutions d’équations aux dérivées partielles linéaires, Rev. Roumaine Math. Pures Appl. 28 (1983), no. 3, 205–228 (French). MR 705453
- Donoghue W.F., “Distributions and Fourier Transforms”, 1969, Academic Press.
- Richard Durrett, Probability: theory and examples, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR 1609153
- Philip M. Morse and Herman Feshbach, Methods of theoretical physics. 2 volumes, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0059774
- Gerald B. Folland, Introduction to partial differential equations, 2nd ed., Princeton University Press, Princeton, NJ, 1995. MR 1357411
- Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
- Anna Karczewska and Jerzy Zabczyk, Stochastic PDE’s with function-valued solutions, Infinite dimensional stochastic analysis (Amsterdam, 1999) Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., vol. 52, R. Neth. Acad. Arts Sci., Amsterdam, 2000, pp. 197–216. MR 1832378
- X. Mao and L. Markus, Wave equation with stochastic boundary values, J. Math. Anal. Appl. 177 (1993), no. 2, 315–341. MR 1231485, DOI 10.1006/jmaa.1993.1261
- Bohdan Maslowski, Stability of semilinear equations with boundary and pointwise noise, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 1, 55–93. MR 1315350
- Annie Millet and Marta Sanz-Solé, A stochastic wave equation in two space dimension: smoothness of the law, Ann. Probab. 27 (1999), no. 2, 803–844. MR 1698971, DOI 10.1214/aop/1022677387
- Annie Millet and Marta Sanz-Solé, Approximation and support theorem for a wave equation in two space dimensions, Bernoulli 6 (2000), no. 5, 887–915. MR 1791907, DOI 10.2307/3318761
- Carl Mueller, Long time existence for the wave equation with a noise term, Ann. Probab. 25 (1997), no. 1, 133–151. MR 1428503, DOI 10.1214/aop/1024404282
- Jacques Neveu, Processus aléatoires gaussiens, Séminaire de Mathématiques Supérieures, No. 34 (Été, vol. 1968, Les Presses de l’Université de Montréal, Montreal, Que., 1968 (French). MR 0272042
- F. Oberhettinger, Tables of Fourier transforms and Fourier transforms of distributions, Springer-Verlag, Berlin, 1990. Translated and revised from the German. MR 1055360, DOI 10.1007/978-3-642-74349-8
- Szymon Peszat, The Cauchy problem for a nonlinear stochastic wave equation in any dimension, J. Evol. Equ. 2 (2002), no. 3, 383–394. MR 1930613, DOI 10.1007/PL00013197
- Szymon Peszat and Jerzy Zabczyk, Nonlinear stochastic wave and heat equations, Probab. Theory Related Fields 116 (2000), no. 3, 421–443. MR 1749283, DOI 10.1007/s004400050257
- Marta Sanz-Solé and Mònica Sarrà, Path properties of a class of Gaussian processes with applications to spde’s, Stochastic processes, physics and geometry: new interplays, I (Leipzig, 1999) CMS Conf. Proc., vol. 28, Amer. Math. Soc., Providence, RI, 2000, pp. 303–316. MR 1803395, DOI 10.1016/s0304-4149(98)00092-1
- Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, IX-X, Hermann, Paris, 1966 (French). Nouvelle édition, entiérement corrigée, refondue et augmentée. MR 0209834
- R. B. Sowers, Multidimensional reaction-diffusion equations with white noise boundary perturbations, Ann. Probab. 22 (1994), no. 4, 2071–2121. MR 1331216
- John B. Walsh, An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour, XIV—1984, Lecture Notes in Math., vol. 1180, Springer, Berlin, 1986, pp. 265–439. MR 876085, DOI 10.1007/BFb0074920
- Richard L. Wheeden and Antoni Zygmund, Measure and integral, Pure and Applied Mathematics, Vol. 43, Marcel Dekker, Inc., New York-Basel, 1977. An introduction to real analysis. MR 0492146
- Calvin H. Wilcox, The Cauchy problem for the wave equation with distribution data: an elementary approach, Amer. Math. Monthly 98 (1991), no. 5, 401–410. MR 1104303, DOI 10.2307/2323855
Additional Information
- Robert C. Dalang
- Affiliation: Institut de Mathématiques, Ecole Polytechnique Fédérale, Station 8, 1015 Lausanne, Switzerland
- Email: robert.dalang@epfl.ch
- Olivier Lévêque
- Affiliation: Institut de Systèmes de Communication, Ecole Polytechnique Fédérale, Station 14, 1015 Lausanne, Switzerland
- Email: olivier.leveque@epfl.ch
- Received by editor(s): January 26, 2004
- Received by editor(s) in revised form: May 4, 2004
- Published electronically: May 9, 2005
- Additional Notes: The research of the first author was partially supported by the Swiss National Foundation for Scientific Research
This article is based on part of the second author’s Ph.D. thesis, written under the supervision of the first author. - © Copyright 2005 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 358 (2006), 2123-2159
- MSC (2000): Primary 60H15; Secondary 60G15, 35R60
- DOI: https://doi.org/10.1090/S0002-9947-05-03740-2
- MathSciNet review: 2197451