### 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 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

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*Journal of Mathematical Physics*, vol. 38, no. 5, pp. 2660-2675. https://doi.org/10.1063/1.531980

**An asymptotic limit law with a singularly perturbed drift and a random noise.** / Kim, Jeong Hoon.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An asymptotic limit law with a singularly perturbed drift and a random noise

AU - Kim, Jeong Hoon

PY - 1997/5

Y1 - 1997/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0031533395&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0031533395&partnerID=8YFLogxK

U2 - 10.1063/1.531980

DO - 10.1063/1.531980

M3 - Article

AN - SCOPUS:0031533395

VL - 38

SP - 2660

EP - 2675

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 5

ER -