Fluctuations around a homogenised semilinear random PDE
File(s)
Author(s)
Hairer, M
Pardoux, E
Type
Journal Article
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.
Date Issued
2020-10-06
Online Publication Date
2021-05-17T10:56:43Z
Date Acceptance
2020-09-14
ISSN
0003-9527
Publisher
Springer
Start Page
151
End Page
217
Journal / Book Title
Archive for Rational Mechanics and Analysis
Volume
239
Copyright Statement
This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any
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Commons licence, unless indicated otherwise in a credit line to the material. If material is not
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licenses/by/4.0/.
medium or format, as long as you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative
Commons licence, unless indicated otherwise in a credit line to the material. If material is not
included in the article’s Creative Commons licence and your intended use is not permitted by
statutory regulation or exceeds the permitted use, you will need to obtain permission directly
from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/
licenses/by/4.0/.
License URI
Sponsor
The Leverhulme Trust
Commission of the European Communities
The Royal Society
Identifier
https://link.springer.com/article/10.1007/s00205-020-01574-8
Grant Number
RL-2012-020-Transfer In
615897
RP/R1/191065
Subjects
Science & Technology
Physical Sciences
Technology
Mathematics, Applied
Mechanics
Mathematics
WONG-ZAKAI THEOREM
Science & Technology
Physical Sciences
Technology
Mathematics, Applied
Mechanics
Mathematics
WONG-ZAKAI THEOREM
math.PR
math.PR
math.AP
60H15, 60H30, 60L30
0101 Pure Mathematics
0102 Applied Mathematics
General Physics
Publication Status
Published
Date Publish Online
2021-10-06