Fluctuations around a homogenised semilinear random PDE

TL;DR

This paper investigates the fluctuations around a homogenised solution of a semilinear random PDE with oscillating potential, revealing different limiting behaviors in dimensions 1, 2, and 3, using advanced regularity structures theory.

Contribution

It provides a detailed analysis of the central limit theorem for semilinear PDEs with random potentials across different dimensions, employing modern regularity structures methodology.

Findings

In 1D, the rescaled difference converges to an Ornstein-Uhlenbeck process.

In 2D, the limit is a non-centred Gaussian process.

In 3D, additional deterministic correction is needed before the CLT limit.

Abstract

We consider a semilinear parabolic partial differential equation in $\mathbf{R}_+\times [0,1]^d$, where $d=1, 2$ or $3$, with a highly oscillating random potential and either homogeneous Dirichlet or Neumann boundary condition. If the amplitude of the oscillations has the right size compared to its typical spatiotemporal scale, then the solution of our equation converges to the solution of a deterministic homogenised parabolic PDE, which is a form of law of large numbers. Our main interest is in the associated central limit theorem. Namely, we study the limit of a properly rescaled difference between the initial random solution and its LLN limit. In dimension $d=1$, that rescaled difference converges as one might expect to a centred Ornstein-Uhlenbeck process. However, in dimension $d=2$, the limit is a non-centred Gaussian process, while in dimension $d=3$, before taking the CLT limit, we need to subtract at an intermediate scale the solution of a deterministic parabolic PDE, subject (in the case of Neumann boundary condition) to a non-homogeneous Neumann boundary condition. Our proofs make use of the theory of regularity structures, in particular of the very recently developed methodology allowing to treat parabolic PDEs with boundary conditions within that theory.