Random nonlinear wave equations: Smoothness of the solutions. (English) Zbl 0635.60073

We show existence and uniqueness for the solution of a one-dimensional wave equation with nonlinear random forcing. Then we give sufficient conditions for the solution at a given time and a given point, to have a density and for this density to be smooth. The proof uses the extension of the Malliavin calculus to two parameters Wiener functionals.
Reviewer: R.Carmona


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G60 Random fields
Full Text: DOI


[1] Bismut, S.M.: Martingales, the Malliavin calculus and hypoellipticity under general Hormander’s conditions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 56, 469-505 (1981) · Zbl 0445.60049
[2] Bouleau, N., Hirsch, F.: Proprietés d’ absolue continuité dans les espaces de Dirichlet et applications aux équations differentielles stochastiques. In: Azéma, J., Yor, M. (eds.) Séminaire de probabilités XX, 1984/85. (Lect. Notes Math., vol. 1204, pp. 131-161) Berlin Heidelberg New York: Springer 1986
[3] Cabana, E.M.: The vibrating string forced by white noise. Z. Wahrscheinlichkeitstheor. Verw. Geb 15, 111-130 (1970) · Zbl 0193.45101
[4] Cabana, E.M.: On barrier problems for the vibrating string. Z. Wahrscheinlichkeitstheor. Verw. Geb. 22, 13-24 (1972) · Zbl 0227.60038
[5] Cairoli, R., Walsh, J.B.: Stochastic integrals in the plane. Acta Math. 134, 111-183 (1975) · Zbl 0334.60026
[6] Carmona, R., Nualart, D.: Random non-linear wave equations: propagations of singularities. Ann. Probab. (in press) · Zbl 0643.60045
[7] Funaki, T.: Random motion of strings and related stochastic evolution equations. Nagoya Math. J. 89, 129-193 · Zbl 0531.60095
[8] Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam New York: North Holland 1981 · Zbl 0495.60005
[9] Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. Part 1. In: (eds.) Stochastic Analysis. Proceedings, Katata 1982. Amsterdam New York: North Holland 1986 · Zbl 0568.60059
[10] Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. Proceedings International Conference on Stochastic Differential Equations, Kyoto, 1976. New York: Wiley 1978
[11] Norris, J.: Simplified Malliavin calculus. In: Azéma, J., Yor, M. (eds.) Seminaire de probabilités XX, 1984/85. (Lect. Notes Math., vol. 1204, pp. 101-130) Berlin Heidelberg New York: Springer 1986
[12] Nualart, D.: Some remarks on a linear stochastic differential equation. Stat. Probab. Lett. (in press) · Zbl 0613.60054
[13] Nualart, D., Sanz, M.: Malliavin Calculus for two-parameter Wiener functionals. Z. Wahrscheinlichkeitstheor. Verw. Geb. 70, 573-590 (1985) · Zbl 0595.60065
[14] Nualart, D., Sanz, M.: Stochastic differential equations on the plane: smoothness of the solution. (preprint) · Zbl 0692.60044
[15] Nualart, D., Zakai, M.: Generalized stochastic integrals and the Malliavin calculus. Prob. Th. Rel. Fields 73, 255-280 (1986) · Zbl 0601.60053
[16] Orsingher: Ann. Inst. Henri Poincare, Nouv. Ser. Sect. B18, 367-394 (1982)
[17] Orsingher, E.: Damped vibrations excited by white noise. Adv. Appl. Probab. 16, 562-584 (1984) · Zbl 0547.60065
[18] Shigekawa, I.: Derivatives of Wiener functionals and absolute continuity of induced measures. J. Math. Kyoto Univ. 20, 263-289 (1980) · Zbl 0476.28008
[19] Stroock, D.: The Malliavin calculus, a functional analytic approach. J. Funct. Anal. 44, 212-257 (1981) · Zbl 0475.60060
[20] Stroock, D.: Some applications of stochastic calculus to partial differential equations. In: Hennequin, P.L. (ed.) Ecole d’Eté de Probabilités de Saint Flour XI. (Lect. Notes Math., vol. 976, pp. 267-382) Berlin Heidelberg New York: Springer 1983
[21] Walsh, J.B.: An introduction to stochastic partial differential equations. In: Carmona, R., Kesten, H., Walsh, J.B. (eds.) Ecole d’Eté de Probabilités de Saint Flour XIV. (Lect. Notes Math., vol. 1180, pp. 266-437) Berlin Heidelberg New York: Springer 1986 · Zbl 0608.60060
[22] Watanabe, S.: Lectures on stochastic differential equations and Malliavin calculus. Berlin Heidelberg New York: Springer 1984 · Zbl 0546.60054
[23] Wong, E., Zakai, M.: Markov processes on the plane. Stochastics 15, 311-333 (1985) · Zbl 0588.60043
[24] Wong, E., Zakai, M.: Multiparameter martingale differential forms. Mem. M84/105 Electron Res. Lab. University of California, Berkeley · Zbl 0612.60045
[25] Zakai, M.: The Malliavin calculus. Acta Appl. Math. 3, 175-207 (1985) · Zbl 0553.60053
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.