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

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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 languageEnglish
Pages (from-to)2660-2675
Number of pages16
JournalJournal of Mathematical Physics
Volume38
Issue number5
DOIs
Publication statusPublished - 1997 May

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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