### Abstract

We consider a one-dimensional linear wave equation with a small mean zero dissipative field and with the boundary condition imposed by the so-called Goursat problem. In order to observe the effect of the randomness on the solution we perform a space-time rescaling and we rewrite the problem in a diffusion approximation form for two parameter processes. We prove that the solution converges in distribution toward the solution of a two-parameter stochastic differential equation which we identify. The diffusion approximation results for oneparameter processes are well known and well understood. In fact, the solution of the one-parameter analog of the problem we consider here is immediate. Unfortunately, the situation is much more complicated for two-parameter processes and we believe that our result is the first one of its kind.

Original language | English (US) |
---|---|

Pages (from-to) | 277-298 |

Number of pages | 22 |

Journal | Probability Theory and Related Fields |

Volume | 98 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 1994 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

## Fingerprint Dive into the research topics of 'A diffusion approximation result for two parameter processes'. Together they form a unique fingerprint.

## Cite this

*Probability Theory and Related Fields*,

*98*(3), 277-298. https://doi.org/10.1007/BF01192255