### Abstract

We show existence and uniqueness for the solution of a onedimensional wave equation with non-linear 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 the two parameters Wiener functionals.

Original language | English (US) |
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Pages (from-to) | 469-508 |

Number of pages | 40 |

Journal | Probability Theory and Related Fields |

Volume | 79 |

Issue number | 4 |

DOIs | |

State | Published - Nov 1 1988 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Analysis
- Mathematics(all)

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## Cite this

Carmona, R. A., & Nualart, D. (1988). Random non-linear wave equations: Smoothness of the solutions.

*Probability Theory and Related Fields*,*79*(4), 469-508. https://doi.org/10.1007/BF00318783