We consider a semilinear parabolic partial differential equation in $$\mathbf{R}_+\times [0,1]^d$$ R + × [ 0 , 1 ] d , where $$d=1, 2$$ 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 $$d=1$$ d = 1 , that rescaled difference converges as one might expect to a centred Ornstein–Uhlenbeck process. However, in dimension $$d=2$$ d = 2 , the limit is a non-centred Gaussian process, while in dimension $$d=3$$ 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.

中文翻译：

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围绕均质半线性随机偏微分方程的波动

我们考虑 $$\mathbf{R}_+\times [0,1]^d$$ R + × [ 0 , 1 ] d 中的半线性抛物线偏微分方程，其中 $$d=1, 2$$ d = 1、2 或 3，具有高度振荡的随机势和齐次 Dirichlet 或 Neumann 边界条件。如果振荡的幅度与其典型的时空尺度相比具有正确的大小，那么我们方程的解就会收敛到确定性均质抛物线偏微分方程的解，这是一种大数定律的形式。我们的主要兴趣是相关的中心极限定理。也就是说，我们研究了初始随机解与其 LLN 限制之间适当重新调整的差异的限制。在维度 $$d=1$$ d = 1 中，重新调整后的差异正如人们所期望的那样收敛到一个中心的 Ornstein-Uhlenbeck 过程。但是，在维度 $$d=2$$ d = 2 中，极限是一个非中心的高斯过程，而在维度 $$d=3$$ d = 3 ，在取 CLT 极限之前，我们需要在中间尺度上减去确定性抛物线偏微分方程的解，主题（在Neumann 边界条件的情况）到非齐次 Neumann 边界条件。我们的证明利用了规则结构理论，特别是最近开发的方法，允许在该理论内处理具有边界条件的抛物线偏微分方程。