Random homogenisation of a highly oscillatory singular potential

In this article, we consider the problem of homogenising the linear heat equation perturbed by a rapidly oscillating random potential. We consider the situation where the space-time scaling of the potential’s oscillations is not given by the diffusion scaling that leaves the heat equation invariant. Instead, we treat the case where spatial oscillations are much faster than temporal oscillations. Under suitable scaling of the amplitude of the potential, we prove convergence to a deterministic heat equation with constant potential, thus completing the results previously obtained in Pardoux and Piatnitski (Ann Probab, 40(3):1316–1356, 2012). © Springer Science+Business Media New York 2013.