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This paper concerns the macroscopic behavior of solutions to parabolicequations with large, highly oscillatory, random potential. When thecorrelation function of the random potential satisfies a specific integrabilitycondition, we show that the random solution converges, as the correlationlength of the medium tends to zero, to the deterministic solution of ahomogenized equation in dimension $d\geq3$. Our derivation is based on aFeynman-Kac probabilistic representation and the Kipnis-Varadhan method appliedto weak convergence of Brownian motions in random sceneries. For sufficientlymixing coefficients, we also provide an optimal rate of convergence to thehomogenized limit using a quantitative martingale central limit theorem. Assoon as the above integrability condition fails, the solution is expected toremain stochastic in the limit of a vanishing correlation length. For a largeclass of potentials given as functionals of Gaussian fields, we show theconvergence of solutions to stochastic partial differential equations (SPDE)with multiplicative noise. The Feynman-Kac representation and the correspondingweak convergence of Brownian motions in random sceneries allows us to explainthe transition from deterministic to stochastic limits as a function of thecorrelation function of the random potential.

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