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In this note we investigate the asymptotic behavior of the solutions of the heat equation with random, fast oscillating potential
∂tuε(t,x)uε(0,x)==12Δxuε(t,x)+ε−γV(xε)uε(t,x),(t,x)∈(0,+∞)×Rd,u0(x),x∈Rd,
(1)
as ε→0 +. We assume that d ≥ 1. The field {V(x),x∈Rd}
is a zero mean, stationary Gaussian random field whose covariance function is given by R(x)=∫Rdeip⋅xa(p)|p|2−2α−ddp, where a(·) is a compactly supported, even, bounded measurable function, continuous at 0 such that a(0) > 0 and α < 1. One can show that then R(x)~C|x|2α − 2 for |x| ≫ 1 where C > 0. It has been shown earlier, see e.g. Lejay (Probab Theory Relat Fields 120:255–276, 2001), that when covariance decays sufficiently fast, i.e. α < 0 and γ = 1 the solutions of Eq. 1 converge to a solution of a certain constant coefficient parabolic (“homogenized”) equation. The situation changes when the decay of correlations is slower, i.e. when α ∈ (0,1). We prove that in that case convergence holds when γ = 1 − α and the appropriate limit is no longer deterministic. It can be described as a solution of the heat equation with a singular potential given by a Gaussian, distribution valued, random field.
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