A Central Limit Theorem for Diffusions with Periodic Coefficients

It is proved that if Xₜ is a diffusion generated by the operator L = 1/2 ∑ aᵢⱼ(x)∂²/∂ xᵢ∂ xⱼ + ∑ u₀bᵢ(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.