Fractional Homogenization of Parabolic Equations with Long-Range Random Potentials

TL;DR

This paper develops a homogenization theory for one-dimensional parabolic equations with long-range correlated random potentials, revealing a stochastic limit described by fractional Gaussian fields and demonstrating superdiffusive behavior.

Contribution

It introduces a novel homogenization framework for equations with non-integrable covariance, showing convergence to fractional Gaussian fields and establishing quantitative convergence rates.

Findings

Solution converges to a fractional Gaussian field with Hurst index > 1/2

Quantitative convergence rate in Wasserstein distance is proportional to ^{ ext{min}(\u03b1,1-rac{1}{4})}

Rescaled fluctuations follow a central limit theorem with ^{-lpha/4} scaling

Abstract

This paper establishes a complete homogenization theory for the one-dimensional parabolic equation with long-range correlated random potential: \[ \partial_t u_\varepsilon(t,x) = \frac{1}{2} \partial_{xx} u_\varepsilon(t,x) + \varepsilon^{-\alpha/2} a\left(\frac{x}{\varepsilon}\right) u_\varepsilon(t,x), \] 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 $\varepsilon^{-\alpha/2}$, the solution converges in distribution to a stochastic limit described by a fractional Gaussian field with Hurst index $H = 1-\alpha/2 > 1/2$: \[ u(t,x) = \mathbb{E}^B\left[\varphi(x+B_t) \exp\left(\beta\int_{\mathbb{R}} L_t^x(y) dW^H(y)\right)\right], \] 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 $W_2(u_\varepsilon, u) \leq C\varepsilon^{\min(\alpha,1-\alpha)/4}$, (iii) a central limit theorem for rescaled fluctuations with scaling $\varepsilon^{-\alpha/4}$, 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.