Fractional Homogenization of Parabolic Equations with Long-Range Random Potentials

Abstract

This paper establishes a complete homogenization theory for the one-dimensional parabolic equation with long-range correlated random potential: \[ \partial_t u_\varepsilon(t,x) = \frac{1}{2} \partial_{xx} u_\varepsilon(t,x) + \varepsilon^{-α/2} a\left(\frac{x}{\varepsilon}\right) u_\varepsilon(t,x), \] where the random field $a$ has covariance decaying as $|x{|}^{-\alpha }$ with $\alpha \in (0,1)$. Contrary to classical homogenization where rapid decorrelation leads to deterministic limits, the non-integrable covariance preserves macroscopic randomness. We prove that under the critical scaling ${\epsilon }^{-\alpha /2}$, the solution converges in distribution to a stochastic limit described by a fractional Gaussian field with Hurst index $H=1-\alpha /2>1/2$: \[ u(t,x) = \mathbb{E}^B\left[φ(x+B_t) \exp\left(β\int_{\mathbb{R}} L_t^x(y) dW^H(y)\right)\right], \] where $W^H$ is fractional Brownian motion and the integral is a Young integral. Our contributions include: (i) functional convergence of the integrated potential to fBm, (ii) quantitative convergence rates in Wasserstein distance $W_2(u_{\epsilon },u)\le C{\epsilon }^{\min (\alpha ,1-\alpha )/4}$, (iii) a central limit theorem for rescaled fluctuations with scaling ${\epsilon }^{-\alpha /4}$, and (iv) superdiffusive transport $\mathbb{E}[X_t^2]\sim t^{2H}$. The results reveal a new homogenization mechanism driven by long-range dependence, connecting stochastic homogenization, fractional calculus, and anomalous diffusion theory.