Random homogenisation of a highly oscillatory singular potential

Martin Hairer , Etienne Pardoux (1) , Andrey Piatnitski (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 LPI RAS - P. N. Lebedev Physical Institute of the Russian Academy of Sciences [Moscow]

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 \textit{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 \cite{MR2962093}.

Domaines: Mathématiques [math]; Probabilités [math.PR]

Martin Hairer, Etienne Pardoux, Andrey Piatnitski. Random homogenisation of a highly oscillatory singular potential. Stochastics and Partial Differential Equations: Analysis and Computations, 2013, 1 (4), pp.571-605. ⟨hal-01231990⟩