Fractional Homogenization of Parabolic Equations with Long-Range Random Potentials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914189960806400 |
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| author | Lechiheb, Atef |
| author_facet | Lechiheb, Atef |
| contents | 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 $|x|^{-α}$ with $α\in (0,1)$. 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 $\varepsilon^{-α/2}$, the solution converges in distribution to a stochastic limit described by a fractional Gaussian field with Hurst index $H = 1-α/2 > 1/2$: \[ u(t,x) = \mathbb{E}^B\left[φ(x+B_t) \exp\left(β\int_{\mathbb{R}} L_t^x(y) dW^H(y)\right)\right], \] 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 $W_2(u_\varepsilon, u) \leq C\varepsilon^{\min(α,1-α)/4}$, (iii) a central limit theorem for rescaled fluctuations with scaling $\varepsilon^{-α/4}$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_08496 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fractional Homogenization of Parabolic Equations with Long-Range Random Potentials Lechiheb, Atef Probability 60H15, 35R60, 60G22, 60F05, 35B27 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 $|x|^{-α}$ with $α\in (0,1)$. 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 $\varepsilon^{-α/2}$, the solution converges in distribution to a stochastic limit described by a fractional Gaussian field with Hurst index $H = 1-α/2 > 1/2$: \[ u(t,x) = \mathbb{E}^B\left[φ(x+B_t) \exp\left(β\int_{\mathbb{R}} L_t^x(y) dW^H(y)\right)\right], \] 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 $W_2(u_\varepsilon, u) \leq C\varepsilon^{\min(α,1-α)/4}$, (iii) a central limit theorem for rescaled fluctuations with scaling $\varepsilon^{-α/4}$, 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. |
| title | Fractional Homogenization of Parabolic Equations with Long-Range Random Potentials |
| topic | Probability 60H15, 35R60, 60G22, 60F05, 35B27 |
| url | https://arxiv.org/abs/2512.08496 |