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A Central Limit Theorem for Diffusions with Periodic Coefficients
The Annals of Probability
Vol. 13, No. 2 (May, 1985), pp. 385-396
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2243798
Page Count: 12
You can always find the topics here!Topics: Brownian motion, Central limit theorem, Solutes, Porous materials, Markov processes, Velocity, Mathematical integrals, Liquids, Coefficients, Transition probabilities
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It is proved that if Xt is a diffusion generated by the operator L = 1/2 ∑ aij(x)∂2/∂ xi∂ xj + ∑ u0b i(x)∂/∂ xi having periodic coefficients, then λ-1/2(Xλ t - λ u0 b̄t), t ≥ 0, converges in distribution to a Brownian motion as λ → ∞. Here b̄ is the mean of b(x) = (b1(x), ⋯, bk(x)) with respect to the invariant distribution for the diffusion induced on the torus Tk = [ 0, 1)k. The dispersion matrix of the limiting Brownian motion is also computed. In case b̄ = 0 this result was obtained by Bensoussan, Lions and Papanicolaou (1978). (See Theorem 4.3, page 401, as well as the author's remarks on page 529.) The case b̄ ≠ 0 is of interest in understanding how solute dispersion in a porous medium behaves as the liquid velocity increases in magnitude.
The Annals of Probability © 1985 Institute of Mathematical Statistics