Fluctuations Around a Homogenised Semilinear Random PDE
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Fluctuations Around a Homogenised Semilinear Random PDE Martin Hairer
& Étienne Pardoux
Communicated by F. Otto
Abstract We consider a semilinear parabolic partial differential equation in R+ ×[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.
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . 2. Assumptions on the Noise . . . . . . . . . . . . . 2.1. Coalescence Trees . . . . . . . . . . . . . . . 2.2. Justification . . . . . . . . . . . . . . . . . . 3. Law of Large Numbers . . . . . . . . . . . . . . . 4. Central Limit Theorem . . . . . . . . . . . . . . . 4.1. Decomposition of the Solution . . . . . . . . 4.2. Definition of the Ambient Regularity Structure 4.3. Description of the Models . . . . . . . . . . . (0) 4.4. Description of vε . . . . . . . . . . . . . .
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M. Hairer & É. Pardoux (1)
. . . . . . . . . . . 4.5. Description of vε 4.6. Formulation of the Fixed Point Problem 5. Convergence of Models . . . . . . . . . . . 5.1. Cumulant Homogeneity Assignments . 5.2. Power-Counting Conditions . . . . . . . 5.3. Special Bounds . . . . . . . . . . . . . 5.4. Tightness and Convergence for the Noise 5.5. Boundary Term . . . . . . . . . . . . . 5.6. Bounds on the Function Q . . . . . . . 6. Auxiliary Results . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .
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