Weak Convergence Approach for Parabolic Equations with Large, Highly Oscillatory, Random Potential

Yu Gu; Guillaume Bal

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arXiv · 1304.5005

Weak Convergence Approach for Parabolic Equations with Large, Highly Oscillatory, Random Potential

Yu Gu·Guillaume Bal

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Abstract

This paper concerns the macroscopic behavior of solutions to parabolic equations with large, highly oscillatory, random potential. When the correlation function of the random potential satisfies a specific integrability condition, we show that the random solution converges, as the correlation length of the medium tends to zero, to the deterministic solution of a homogenized equation in dimension $d\geq3$. Our derivation is based on a Feynman-Kac probabilistic representation and the Kipnis-Varadhan method applied to weak convergence of Brownian motions in random sceneries. For sufficiently mixing coefficients, we also provide an optimal rate of convergence to the homogenized limit using a quantitative martingale central limit theorem. As soon as the above integrability condition fails, the solution is expected to remain stochastic in the limit of a vanishing correlation length. For a large class of potentials given as functionals of Gaussian fields, we show the convergence of solutions to stochastic partial differential equations (SPDE) with multiplicative noise. The Feynman-Kac representation and the corresponding weak convergence of Brownian motions in random sceneries allows us to explain the transition from deterministic to stochastic limits as a function of the correlation function of the random potential.

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BibTeX RIS

Yu Gu, Guillaume Bal. 2014-08-04. Weak Convergence Approach for Parabolic Equations with Large, Highly Oscillatory, Random Potential. https://arxiv.org/abs/1304.5005

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