Fractional Homogenization of Parabolic Equations with Long-Range Random Potentials

This paper establishes a complete homogenization theory for the one-dimensional parabolic equation with long-range correlated random potential: $${\mathrm{\partial }}_t{u}_{\epsilon }(t,x)=\frac{1}{2}{\mathrm{\partial }}_{xx}{u}_{\epsilon }(t,x)+{\epsilon }^{-\alpha \mathrm{/}2}a\left(\frac{x}{\epsilon }\right)u_{\epsilon }(t,x),$$ where the random field $a$ has covariance decaying as $\mathrm{\mid }x{\mathrm{\mid }}^{-\alpha }$ 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 ${\epsilon }^{-\alpha \mathrm{/}2}$, the solution converges in distribution to a stochastic limit described by a fractional Gaussian field with Hurst index $H=1-\alpha \mathrm{/}2>1\mathrm{/}2$: $$u(t,x)={\mathbb{E}}^B[\phi (x+B_t)\exp (\beta {\int }_{\mathbb{R}}{L}_t^x(y)d{W}^H(y))],$$ 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 $W_2(u_{\epsilon },u)\le C{\epsilon }^{\min (\alpha ,1-\alpha )\mathrm{/}4}$, (iii) a central limit theorem for rescaled fluctuations with scaling ${\epsilon }^{-\alpha \mathrm{/}4}$, 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.