Fluctuations Around a Homogenised Semilinear Random PDE

Area of Science:

• Stochastic Partial Differential Equations
• Mathematical Physics
• Homogenization Theory

Background:

• Investigating the behavior of solutions to semilinear parabolic PDEs with highly oscillating random potentials.
• Understanding the convergence of random solutions to deterministic homogenized PDEs (Law of Large Numbers).
• Analyzing the fluctuations around the homogenized limit (Central Limit Theorem).

Purpose of the Study:

• To establish the Central Limit Theorem for semilinear parabolic PDEs with random potentials in dimensions 1, 2, and 3.
• To characterize the limiting random processes for the rescaled differences between random and homogenized solutions.
• To explore the impact of dimensionality and boundary conditions on the homogenization and fluctuation analysis.

Main Methods:

• Application of the theory of regularity structures, specifically tailored for parabolic PDEs with boundary conditions.
• Analysis of homogenization via a Law of Large Numbers for the PDE solutions.
• Study of the Central Limit Theorem by examining the limit of rescaled solution differences.
• Utilizing recently developed methodologies within regularity structures for boundary value problems.

Main Results:

• In dimension 1, the rescaled difference converges to a centered Ornstein-Uhlenbeck process.
• In dimension 2, the limit is a non-centered Gaussian process.
• In dimension 3, an intermediate deterministic parabolic PDE solution with a non-homogeneous Neumann condition must be subtracted before applying the CLT.

Conclusions:

• The study provides a comprehensive analysis of homogenization and fluctuations for parabolic PDEs with random potentials.
• The limiting behavior of the fluctuations depends significantly on the spatial dimension.
• The theory of regularity structures offers a powerful framework for analyzing such complex PDEs with boundary conditions.