On the Asymptotic Behavior of Solutions of the Heat Equation with a Random, Long-Range Correlated Potential

Abstract

In this note we investigate the asymptotic behavior of the solutions of the heat equation with random, fast oscillating potential

$$\begin{array}{rcl} \partial_tu_{\varepsilon}(t,x)&=&\dfrac12\Delta_xu_{\varepsilon}(t,x)+{\varepsilon}^{-\gamma}V\left(\dfrac{x}{{\varepsilon}}\right)u_{\varepsilon}(t,x),\,(t,x)\in(0,+\infty)\times{\mathbb R}^d, \\ u_{\varepsilon}(0,x)&=&u_0(x),\,x\in{\mathbb R}^d, \end{array} $$

(1)

as ε→0 +. We assume that d ≥ 1. The field $\{V(x),\,x\in{\mathbb R}^d\}$ is a zero mean, stationary Gaussian random field whose covariance function is given by $R(x)=\int_{{\mathbb R}^d} e^{ip\cdot x}a(p)|p|^{2-2{\alpha}-d}dp$, 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.