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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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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DOI: https://doi.org/10.1007/s004400100159
Keywords
- Asymptotic Behavior
- Random Field
- Heat Equation
- Constant Coefficient
- Functional Space