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Diffusion approximation for the convection-diffusion equation with random drift
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  • Published: December 2001

Diffusion approximation for the convection-diffusion equation with random drift

  • Tomasz Komorowski1 

Probability Theory and Related Fields volume 121, pages 525–550 (2001)Cite this article

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Abstract.

We consider the asymptotic behavior of the solutions ofscaled convection-diffusion equations ∂ t u ɛ (t, x) = κΔ x (t, x) + 1/ɛV(t/ɛ2,xɛ) ·∇ x u ɛ (t, x) with the initial condition u ɛ(0,x) = u 0(x) as the parameter ɛ↓ 0. Under the assumptions that κ > 0 and V(t, x), (t, x) ∈R d is a d-dimensional,stationary, zero mean, incompressible, Gaussian random field, Markovian and mixing in t we show that the laws of u ɛ(t,·), t≥ 0 in an appropriate functional space converge weakly, as ɛ↓ 0, to a δ-type measureconcentrated on a solution of a certain constant coefficient heat equation.

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Authors and Affiliations

  1. Mathematical Institute of the Polish Academy of Sciences, Warsaw, Institute of Mathematics, UMCS, Lublin, Poland. e-mail: komorow@golem.umcs.lublin.pl., , , , , , PL

    Tomasz Komorowski

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  1. Tomasz Komorowski
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Received: 23 March 2000 / Revised version: 5 March 2001 / Published online: 9 October 2001

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Komorowski, T. Diffusion approximation for the convection-diffusion equation with random drift. Probab Theory Relat Fields 121, 525–550 (2001). https://doi.org/10.1007/s004400100159

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  • Issue Date: December 2001

  • DOI: https://doi.org/10.1007/s004400100159

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Keywords

  • Asymptotic Behavior
  • Random Field
  • Heat Equation
  • Constant Coefficient
  • Functional Space
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