### Abstract

An asymptotic stochastic initial value problem with a random microscale superposed upon a deterministic macroscale is considered in this article. We derive and prove a limit theorem for the random problem with a rapidly varying deterministic component. The asymptotic character of the stochastic initial value problem with a small parameter is realized by solving a final value problem of which the infinitesimal generator consists of a singularly perturbed deterministic component and a random fluctuation intensity component. We also give an estimate for the error in the asymptotic approximation in terms of the small parameter. This abstract limit theorem is reduced to a limit theorem for the stochastic processes solving a system of stochastic differential equations. The corresponding infinitesimal generator of the Kolmogorov-Fokker-Planck equation is obtained in an asymptotic form and demonstrates how an effective driving force couples with a zero-mean random perturbation in both drift and diffusion coefficients. Our theorem provides a characterization of random noise for evanescent waves in a layered random medium.

Original language | English |
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Pages (from-to) | 2660-2675 |

Number of pages | 16 |

Journal | Journal of Mathematical Physics |

Volume | 38 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1997 May |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics