Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Periodic homogenisation for two dimensional generalised parabolic Anderson model

Yilin Chen, Benjamin Fehrman, Weijun Xu

Abstract

We consider the periodic homogenisation problem for the generalised parabolic Anderson model on two dimensional torus. We show that, for the renormalisation that respects Wick ordering, the homogenisation and renormalisation procedures commute. The main novelty is to identify a suitable ansatz for the solution on top of the usual para-controlled ansatz to set up a fixed point problem uniform in the homogenisation parameter. After that, one further utilises cancellations and resonances from the homogenisation oscillations to show convergence of both the solution and flux to the right limits. At a technical level, we frequently use integration by parts as well as "completing the products" to circumvent the incompatibility between para-products and variable coefficients. As a byproduct, we also show that the standard two dimensional generalised parabolic Anderson model can be constructed with para-controlled calculus without using commutator estimates.

Periodic homogenisation for two dimensional generalised parabolic Anderson model

Abstract

We consider the periodic homogenisation problem for the generalised parabolic Anderson model on two dimensional torus. We show that, for the renormalisation that respects Wick ordering, the homogenisation and renormalisation procedures commute. The main novelty is to identify a suitable ansatz for the solution on top of the usual para-controlled ansatz to set up a fixed point problem uniform in the homogenisation parameter. After that, one further utilises cancellations and resonances from the homogenisation oscillations to show convergence of both the solution and flux to the right limits. At a technical level, we frequently use integration by parts as well as "completing the products" to circumvent the incompatibility between para-products and variable coefficients. As a byproduct, we also show that the standard two dimensional generalised parabolic Anderson model can be constructed with para-controlled calculus without using commutator estimates.
Paper Structure (16 sections, 37 theorems, 235 equations)

This paper contains 16 sections, 37 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. Overview of the strategy
  3. Uniform boundedness and convergence of the solution
  4. Setting up the fixed point equations
  5. Some preliminary lemmas
  6. Proof of Theorem \ref{['thm:fixed_pt_thm']}
  7. Convergence of the flux
  8. Convergence of the enhanced stochastic terms
  9. The free field
  10. The linear flux and its variants
  11. The Wick ordered term
  12. The mild quadratic term
  13. Bounds on Green's functions
  14. Some preliminary bounds on oscillations and convolutions
  15. Bony's decomposition and para-products
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $\alpha \in (\frac{3}{4},1)$. For every $\varepsilon = \varepsilon_N = \frac{1}{N}$, let $u_\varepsilon$ be the solution to e:gPAM_formal as specified above with initial data $u_\varepsilon (0) \in \mathcal{C}^{\alpha}$. Suppose there exists $u(0)$ such that $u_\varepsilon (0) \rightarrow u(0)$

Theorems & Definitions (74)

  • Theorem 1.2
  • proof
  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 64 more