Random homogenisation of a highly oscillatory singular potential

Abstract

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.

A Class of Growth Models Rescaling To kpz

“…One may ask whether the arguments in this article still hold if ξ is an arbitrary smooth and stationary space-time random field with suitable integrability and mixing conditions. (Think of conditions similar to those considered in [PP12,HPP13]. )…”

Multiplicative stochastic heat equations on the whole space

“…Second, our spaces of modelled distributions are weighted at infinity; therefore, the reconstruction theorem and the abstract convolution with the heat kernel need to be modified in consequence, we refer to Theorems 3.10 and 4.3. One major difficulty we run into is that one would like to consider the same kind of weights as in [HPP13,HL15], which are of the type w(t, x) = exp(t(1 + |x|)). Unfortunately, such weights do not satisfy the very desirable property c ≤ |w(z)/w(z ′ )| ≤ C for some constants c, C > 0, uniformly over space-time points z, z ′ with |z − z ′ | ≤ 1, although they do satisfy this property for pairs of points that are only separated in space.…”

A simple construction of the continuum parabolic Anderson model on $\mathbf{R}^2$

“…The main trick that spares us from using elaborate renormalisation theories is to introduce the "stationary" solution Y of the (additive) stochastic heat equation and to solve the PDE associated to v = ue Y instead of u. This is analogous to what was done for example in [DPD02,HPP13]. The second issue is dealt with by choosing an appropriate time-increasing weight for the solution u.…”